#545715

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1698 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)(13,14,18,16)(15,21,23,19)(17,24,25,20)(22,26,28,27)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(6,9)(10,12)(14,16)(19,21)(20,24)(26,27)(29,30,33,36)(31,34,39,37)(32,35,41,40)(38,42,44,43);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(5,7,8,11)(6,10,9,12)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fh,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 452911);