#7198062

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=fghi,e^2=h,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=dhi

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, 7105258);