#726600

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2500 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)(23,27)(24,28)(25,26)(29,33)(30,35)(31,36)(32,34);(5,6,8,10)(7,9,12,11)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(28,34)(30,31);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(21,23,25,29)(22,27,26,33)(24,31,32,30)(28,36,34,35);(5,7,8,12)(6,9,10,11)(13,15,16,19)(14,17,18,20)(21,24,25,32)(22,28,26,34)(23,30,29,31)(27,35,33,36);(21,22)(23,27)(24,28)(25,26)(29,33)(30,35)(31,36)(32,34)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,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, 633796);