#6697844

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6605040);