#7082912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=h,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6990108);