#323252

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=h,d^2=ehi,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, 230448);