#236270

order	512
nilpotency class	3
isoclinism	isoclinic class of #236265
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

1243 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:

(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,31)(25,36)(30,41)(32,40)(33,39)(35,43)(37,42)(38,45)(44,46)(47,48);(4,13)(7,12)(9,16)(11,14)(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)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(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=i,b^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,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, 143466);