#413824

order	512
nilpotency class	3
isoclinism	isoclinic class of #411625
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 246 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(9,10,15,20)(11,16,23,35)(12,17,26,38)(13,28,31,56)(14,32,29,19)(18,40,41,30)(21,42,44,33)(22,46,48,70)(24,50,47,36)(25,51,53,71)(27,55,52,39)(34,59,60,49)(37,61,62,54)(43,65,67,69)(45,58,66,64)(57,68,63,72);(1,2,4,6)(3,7,8,5)(9,11)(10,16)(12,21)(13,22)(14,24)(15,23)(17,33)(18,34)(19,36)(20,35)(25,43)(26,44)(27,45)(28,46)(29,47)(30,49)(31,48)(32,50)(37,57)(38,42)(39,58)(40,59)(41,60)(51,69)(52,66)(53,67)(54,68)(55,64)(56,70)(61,72)(62,63)(65,71);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,21)(13,25)(14,27)(15,26)(16,33)(18,37)(19,39)(20,38)(22,43)(23,44)(24,45)(28,51)(29,52)(30,54)(31,53)(32,55)(34,57)(35,42)(36,58)(40,61)(41,62)(46,69)(47,66)(48,67)(49,68)(50,64)(56,71)(59,72)(60,63)(65,70);(9,13,29,40,15,31,14,30)(10,18,32,56,20,41,19,28)(11,22,47,59,23,48,24,49)(12,25,52,61,26,53,27,54)(16,34,50,70,35,60,36,46)(17,37,55,71,38,62,39,51)(21,43,66,72,44,67,45,68)(33,57,64,65,42,63,58,69)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321020);