#600404

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

1654 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)(13,14,18,16)(15,21,23,19)(17,24,25,20)(22,26,28,27)(29,30,35,33)(31,38,32,36)(34,37,42,41)(39,44,40,43);(7,12)(9,11)(14,16)(17,25)(19,21)(22,28)(29,31,35,32)(30,36,33,38)(34,40,42,39)(37,44,41,43);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,32,35,31)(30,36,33,38)(34,40,42,39)(37,43,41,44);(5,7)(6,9)(8,12)(10,11)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,33,35,30)(31,38,32,36)(34,41,42,37)(39,44,40,43);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,34)(30,37)(31,39)(32,40)(33,41)(35,42)(36,43)(38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=hi,d^2=f,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, 507600);