#399264

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

995 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)(15,19)(16,30)(17,22)(18,23)(20,31)(21,33)(24,34)(25,41)(26,43)(27,36)(28,37)(29,40)(32,39)(35,42)(38,44);(1,2,4,3)(6,9)(10,12)(13,15,17,27)(14,19,22,36)(16,26,31,44)(18,29,34,28)(20,38,30,43)(21,35,39,41)(23,40,24,37)(25,33,42,32);(1,2,4,3)(13,16)(14,20)(15,25)(17,31)(18,33)(19,35)(21,24)(22,30)(23,39)(26,29)(27,42)(28,44)(32,34)(36,41)(37,43)(38,40);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,26)(17,31)(18,32)(19,35)(20,24)(22,39)(23,30)(25,29)(27,44)(28,42)(33,34)(36,41)(37,38)(40,43)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306460);