#584488

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1634 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6,8,10)(7,11,12,9)(13,14,17,18)(15,21,23,19)(16,24,25,20)(22,26,28,27)(30,34)(32,40)(35,37)(38,44);(7,12)(9,11)(14,18)(19,21)(20,24)(26,27)(29,30,33,34)(31,37,39,35)(32,36,40,41)(38,43,44,42);(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44);(5,7,8,12)(6,9,10,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,33,39)(30,35,34,37)(32,38,40,44)(36,42,41,43);(1,3,4,2)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fh,d^2=fgh,e^2=gi,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 491684);