#6874871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=i,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6782067);