#6815349

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31228	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6722545);