#232910

order	512
nilpotency class	3
isoclinism	isoclinic class of #232905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60961	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1159 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140106);