#718408

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1812 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=gh,e^2=fh,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, 625604);