#7182591

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

1684 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)(8,16)(11,17)(15,20)(18,19)(21,22,27,28,26,31,23,32)(24,34,33,25,30,35,36,29);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,27)(22,28,31,32)(24,33,30,36)(25,34,29,35);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,26,30)(22,29,31,25)(23,33,27,36)(28,35,32,34);(2,4)(7,14)(10,13)(15,20)(18,19)(21,25,26,29)(22,30,31,24)(23,34,27,35)(28,36,32,33);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,31)(25,29)(28,32)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7089787);