#591613

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgi,d^2=fg,e^2=h,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, 498809);