#6491663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398859);