#280868

order	512
nilpotency class	3
isoclinism	isoclinic class of #280843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

466 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,24,30)(19,31,34,27)(20,35,40,46)(21,25,33,42)(22,29,23,28)(26,39,45,38)(32,47,48,41)(36,44,37,43);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,13)(15,16)(17,19,23,33,24,34,22,21)(18,25,29,27,30,42,28,31)(20,36,38,32,40,37,39,48)(26,43,35,47,45,44,46,41);(3,10)(5,13)(6,14)(8,16)(9,15)(11,12)(17,20)(18,26)(19,32)(21,37)(22,38)(23,39)(24,40)(25,41)(27,44)(28,35)(29,46)(30,45)(31,43)(33,36)(34,48)(42,47);(2,7)(6,9)(8,16)(14,15)(17,21,22,34,24,33,23,19)(18,27,28,25,30,31,29,42)(20,37,38,48,40,36,39,32)(26,44,35,41,45,43,46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188064);