#233604

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
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	#60949	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1239 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)(4,12)(8,13)(11,14)(15,16)(17,18)(19,27)(20,24)(21,33)(22,26)(23,30)(25,35)(28,43)(29,41)(31,39)(32,47)(34,37)(36,48)(38,44)(40,46)(42,45);(1,2,5,8)(3,10,11,7)(4,13,12,6)(9,16,14,15)(17,19,22,29)(18,23,26,37)(20,28,31,45)(21,34,35,30)(24,38,39,36)(25,41,33,27)(32,48,46,44)(40,42,47,43);(1,3)(2,7)(4,14)(5,11)(6,15)(8,10)(9,12)(13,16)(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);(3,11)(7,10)(9,14)(15,16)(19,29)(20,31)(23,37)(24,39)(27,41)(30,34)(32,46)(40,47)

Polycyclic presentation:

2^9|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,d^b=dhi,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

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, 140800);