#232845

order	512
nilpotency class	3
isoclinism	isoclinic class of #232834
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	#60962	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140041);