#235441

order	512
nilpotency class	3
isoclinism	isoclinic class of #235438
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1242 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,23,28)(19,29,33,39)(20,34,35,43)(21,27,22,26)(24,32,40,31)(25,37,42,36)(30,41,45,38)(44,47,48,46);(6,15)(9,16)(17,19,23,33)(18,24,28,40)(20,30,35,45)(21,32,22,31)(25,41,42,38)(26,39,27,29)(34,44,43,48)(36,47,37,46);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,38)(26,34)(27,43)(28,42)(29,44)(31,46)(32,47)(33,45)(39,48)(40,41);(20,35)(25,42)(30,45)(34,43)(36,37)(38,41)(44,48)(46,47)

Polycyclic presentation:

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

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, 142637);