#409063

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=fhi,c^2=fi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,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, 316259);