#235137

order	512
nilpotency class	3
isoclinism	isoclinic class of #235117
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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1231 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:

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

Polycyclic presentation:

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

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, 142333);