#232810

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
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	#60688	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1180 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)(3,11)(4,10)(5,9)(6,8)(7,12)(13,15)(14,16)(17,18)(19,28)(20,33)(21,38)(22,27)(23,26)(24,31)(25,37)(29,46)(30,40)(32,41)(34,45)(35,43)(36,44)(39,47)(42,48);(1,3,6,14)(2,7,8,15)(4,13,5,12)(9,16,10,11)(17,19,23,30)(18,24,26,41)(20,34,35,48)(21,32,22,31)(25,42,44,45)(27,40,38,28)(29,37,47,36)(33,39,43,46);(1,2)(3,11)(4,10)(5,9)(6,8)(7,12)(13,15)(14,16)(17,20)(18,25)(19,29)(21,36)(22,37)(23,35)(24,39)(26,44)(27,43)(28,45)(30,47)(31,34)(32,48)(33,38)(40,42)(41,46);(2,8)(7,15)(9,10)(11,16)(18,26)(19,30)(20,35)(27,38)(31,32)(36,37)(39,46)(42,45)

Polycyclic presentation:

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

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, 140006);