#720108

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2202 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)(5,6,10,8)(7,11,9,12)(15,16)(18,19)(21,22)(23,29)(24,32)(25,26)(27,30)(28,34)(31,35)(33,36);(5,7,10,9)(6,11,8,12)(13,14,17,20)(15,18,16,19)(22,26)(23,30)(28,32)(31,33);(5,8,10,6)(7,11,9,12)(13,15,17,16)(14,18,20,19)(21,23)(22,27)(24,33)(25,30)(26,29)(28,36)(31,34)(32,35);(1,2,3,4)(5,9,10,7)(6,11,8,12)(13,16,17,15)(14,19,20,18)(21,24)(22,28)(23,31)(25,34)(26,32)(27,35)(29,36)(30,33);(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,34)(33,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=gh,d^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 627304);