#637033

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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

1988 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)(7,11)(10,12)(13,14,17,20)(15,18,16,19)(21,22)(23,24)(25,33)(26,27)(28,34)(29,32)(30,36)(31,35);(5,6,9,8)(7,10,11,12)(13,14,17,20)(15,18,16,19)(22,27)(23,29)(28,33)(30,35);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,23,26,29)(22,24,27,32)(25,30,34,35)(28,31,33,36);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15,17,16)(14,18,20,19)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,35)(34,36);(5,8,9,6)(7,12,11,10)(13,16,17,15)(14,19,20,18)(21,25,26,34)(22,28,27,33)(23,30,29,35)(24,31,32,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=f,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 544229);