#232895

order	512
nilpotency class	3
isoclinism	isoclinic class of #232889
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18,23,28)(19,29,32,43)(20,33,36,41)(21,27,22,26)(24,37,39,31)(25,35,42,34)(30,38,47,40)(44,45,48,46);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19,23,32)(18,24,28,39)(20,30,36,47)(21,37,22,31)(25,40,42,38)(26,43,27,29)(33,44,41,48)(34,46,35,45);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,33)(27,41)(28,42)(29,44)(31,45)(32,47)(37,46)(39,40)(43,48);(1,4,6,5)(2,8,9,15)(3,12,14,13)(7,10,16,11)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,34,36,35)(24,29,39,43)(25,33,42,41)(30,45,47,46)(38,44,40,48)

Polycyclic presentation:

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

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, 140091);