#547094

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

1706 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)(13,14,18,15)(16,19,24,21)(17,22,25,20)(23,27,28,26)(31,37)(34,40)(38,43)(41,44);(6,9)(10,12)(14,15)(19,21)(20,22)(26,27)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,34)(32,38)(33,37)(35,41)(36,40)(39,43)(42,44);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,32,33,39)(30,35,36,42)(31,38,37,43)(34,41,40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=fg,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, 454290);