#233579

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1261 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)(17,18,22,26)(19,27,30,41)(20,24,31,39)(21,33,35,25)(23,34,37,29)(28,43,44,42)(32,47,46,40)(36,45,38,48);(6,15)(9,16)(17,19,22,30)(18,23,26,37)(20,28,31,44)(21,34,35,29)(24,38,39,36)(25,41,33,27)(32,48,46,45)(40,42,47,43);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20)(18,24)(19,28)(21,32)(22,31)(23,36)(25,40)(26,39)(27,42)(29,45)(30,44)(33,47)(34,48)(35,46)(37,38)(41,43);(20,31)(24,39)(28,44)(32,46)(36,38)(40,47)(42,43)(45,48)

Polycyclic presentation:

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

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, 140775);