#7182396

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,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=dfi,e^b=ehi,d^c=dh,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089592);