#413816

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
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	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,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, 321012);