#620135

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2286 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)(5,6,8,10)(7,11,12,9)(15,19)(17,20)(21,22,25,26)(23,27,29,33)(24,31,32,28)(30,36,35,34);(2,4)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(28,31)(30,35);(1,2,3,4)(5,7,8,12)(6,9,10,11)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(5,6)(7,9)(8,10)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23)(22,27)(24,30)(25,29)(26,33)(28,34)(31,36)(32,35);(21,24,25,32)(22,28,26,31)(23,30,29,35)(27,34,33,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=ghi,c^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

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, 527331);