#6830885

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#42478	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6738081);