#374998

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,15)(13,14)(17,18,22,27,23,28,21,26)(19,24,30,38,31,39,35,43)(20,25,34,42,33,41,36,44)(29,37,46,48,45,47,32,40);(1,3,5,4)(2,6,7,8)(10,14)(15,16)(17,19)(18,24)(20,32)(21,35)(22,30)(23,31)(25,40)(26,43)(27,38)(28,39)(29,34)(33,46)(36,45)(37,42)(41,48)(44,47);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20,22,34,23,33,21,36)(18,25,27,42,28,41,26,44)(19,29,30,46,31,45,35,32)(24,37,38,48,39,47,43,40);(2,7)(6,8)(9,12,13,11)(10,16,14,15)(20,33)(25,41)(29,45)(32,46)(34,36)(37,47)(40,48)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=eh,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 282194);