#751620

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2352 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)(6,9)(10,12)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,34)(35,36);(14,18)(19,20)(22,27)(23,29)(25,33)(31,36);(1,2,3,4)(13,14)(15,19)(16,20)(17,18)(21,23,26,29)(22,24,27,32)(25,30,33,35)(28,31,34,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,16)(14,19,18,20)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,20,18,19)(21,25,26,33)(22,28,27,34)(23,30,29,35)(24,31,32,36)

Polycyclic presentation:

2^9|c^2=fg,d^2=fh,e^2=hi,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, 658816);