#7091115

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#45057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,21,23,24,32,31)(19,25,28,26,27,30,29,20);(9,10)(11,15)(12,14)(13,16)(17,19,23,27)(18,20,24,26)(21,30,31,25)(22,29,32,28);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(11,16)(13,15)(17,21,22,24,23,31,32,18)(19,26,28,30,27,20,29,25);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,27)(20,26)(25,30)(28,29)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=f,e^2=g,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

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, 6998311);