#7188516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1263 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)(3,6)(4,8)(5,7)(10,14)(12,15)(17,18,24,29)(19,26,20,21)(22,28,23,27)(25,31,32,30);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,12,16,15)(17,19,24,20)(18,25,29,32)(21,27,26,28)(22,30,23,31);(2,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20,24,19)(18,26,29,21)(22,30,23,31)(25,28,32,27);(1,4,5,3)(2,8,7,6)(10,14)(11,16)(17,21,24,26)(18,19,29,20)(22,25,23,32)(27,30,28,31);(9,12,13,15)(10,11,14,16)(17,22,24,23)(18,27,29,28)(19,30,20,31)(21,25,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7095712);