#278632

order	512
nilpotency class	3
isoclinism	isoclinic class of #278603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

496 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,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=dh,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185828);