#592002

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

1622 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)(6,7)(8,12)(13,14,18,16)(15,21,23,19)(17,24,25,20)(22,26,28,27)(30,34)(31,37)(32,40)(42,44);(6,7)(10,11)(14,16)(17,25)(19,21)(22,28)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(1,3,4,2)(5,6,9,7)(8,10,12,11)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,31,33,37)(30,35,34,38)(32,39,40,43)(36,42,41,44);(5,8)(6,10)(7,11)(9,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=fgi,d^2=g,e^2=fgh,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, 499198);