#6841197

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31198	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,5)(3,7)(9,10)(11,15)(12,16)(13,14)(18,22)(23,28)(24,29)(30,32);(1,2,4,5)(3,6,7,8)(18,23)(19,25)(21,27)(22,28)(24,29)(26,31);(10,14)(15,16)(17,18,21,23,20,22,27,28)(19,24,25,30,26,29,31,32);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(19,26)(24,29)(25,31)(30,32);(2,5)(6,8)(9,12,13,11)(10,16,14,15)(17,19,21,25,20,26,27,31)(18,24,23,30,22,29,28,32)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=fgi,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313767, #72313904, #72314016
maximal normal subgroup of	#72313767, #72313904, #72314016

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6748393);