#770156

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 677352);