#6718030

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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	#25092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18,21,25)(19,26,30,23)(20,31,22,24)(27,28,32,29);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,21,30)(18,23,25,26)(20,29,22,28)(24,27,31,32);(10,15)(14,16)(17,18)(19,27)(20,24)(21,25)(22,31)(23,28)(26,29)(30,32);(2,6)(5,7)(17,20,21,22)(18,24,25,31)(19,28,30,29)(23,32,26,27);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(19,29)(20,22)(23,32)(24,31)(26,27)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6625226);