#232902

order	512
nilpotency class	3
isoclinism	isoclinic class of #232897
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

1167 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)(3,10)(4,15)(5,9)(6,8)(7,12)(11,14)(13,16)(17,18,23,26)(19,29,32,43)(20,33,34,41)(21,28,22,27)(24,37,39,31)(25,35,42,36)(30,38,45,40)(44,47,48,46);(1,3,6,14)(2,7,8,16)(4,13,5,12)(9,11,15,10)(17,19,23,32)(18,24,26,39)(20,30,34,45)(21,37,22,31)(25,40,42,38)(27,43,28,29)(33,44,41,48)(35,47,36,46);(1,2)(3,11)(4,9)(5,15)(6,8)(7,12)(10,14)(13,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,34)(24,38)(26,42)(27,33)(28,41)(29,44)(31,47)(32,45)(37,46)(39,40)(43,48);(2,8)(7,16)(9,15)(10,11)(18,26)(20,34)(24,39)(27,28)(29,43)(30,45)(35,36)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^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,e^b=eh,g^b=ghi,d^c=dh,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, 140098);