#299199

order	512
nilpotency class	3
isoclinism	isoclinic class of #299195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

907 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 907 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206395);