#232881

order	512
nilpotency class	3
isoclinism	isoclinic class of #232876
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1131 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)(3,10)(4,6)(5,15)(7,9)(8,12)(11,16)(13,14)(17,18,23,28)(19,29,31,43)(20,33,36,41)(21,27,22,26)(24,37,39,32)(25,34,42,35)(30,38,45,40)(44,47,48,46);(1,3)(2,8)(4,14)(5,13)(6,11)(7,12)(9,16)(10,15)(17,19)(18,24)(20,30)(21,37)(22,32)(23,31)(25,40)(26,43)(27,29)(28,39)(33,44)(34,47)(35,46)(36,45)(38,42)(41,48);(1,4,6,2)(3,10,11,16)(5,15,7,9)(8,12,14,13)(17,20,23,36)(18,25,28,42)(19,30,31,45)(21,35,22,34)(24,38,39,40)(26,33,27,41)(29,44,43,48)(32,47,37,46);(3,11)(8,14)(10,16)(12,13)(19,31)(24,39)(29,43)(30,45)(32,37)(38,40)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,c^2=h,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,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140077);