#232826

order	512
nilpotency class	3
isoclinism	isoclinic class of #232820
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1188 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:

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

Polycyclic presentation:

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

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, 140022);