#409082

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^b=gi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316278);