#605624

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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)(7,11)(10,12)(13,14,18,19)(15,21,23,16)(17,25,26,20)(22,24,28,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(6,9)(10,12)(14,19)(16,21)(17,26)(22,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,16,19,21)(17,22,26,28)(20,24,25,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(1,3,4,2)(13,17,18,26)(14,20,19,25)(15,22,23,28)(16,24,21,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fhi,e^2=fgi,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,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, 512820);