#7020056

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	#44886	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6927252);