#7193370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994107
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,9,13)(7,11,14,10)(8,16,18,12)(15,19,17,20)(22,27)(25,34)(28,36)(29,31)(30,35)(32,33);(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,22,26,30)(23,31,24,28)(25,35,34,27)(29,32,36,33);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,23,26,24)(22,28,30,31)(25,32,34,33)(27,36,35,29);(7,14)(10,11)(15,17)(19,20)(21,24,26,23)(22,29,30,36)(25,32,34,33)(27,31,35,28);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(22,30)(27,35)(28,31)(29,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=i,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=dhi,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, 7100566);