#365886

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

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

786 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18,23,28,21,26,22,27)(19,24,31,40,30,39,36,43)(20,32,35,44,34,25,37,42)(29,45,47,41,46,38,33,48);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,36,21,30,23,31)(18,24,27,43,26,39,28,40)(20,33,37,46,34,47,35,29)(25,41,44,45,32,48,42,38);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,21,34)(18,25,26,32)(19,29,30,46)(22,37,23,35)(24,38,39,45)(27,44,28,42)(31,47,36,33)(40,48,43,41);(1,4,5,3)(2,7,6,8)(20,34)(25,32)(29,46)(33,47)(35,37)(38,45)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=ef,d^2=e,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, 273082);