#323231

order	512
nilpotency class	3
isoclinism	isoclinic class of #323197
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,d^2=ei,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,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, 230427);