#6868357

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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)(17,18,22,29,23,28,38,27)(19,30,33,42,34,24,39,43)(20,35,37,45,21,26,36,25)(31,41,48,40,32,46,47,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,34)(18,24,28,30)(20,32,21,31)(22,33,38,39)(25,44,45,40)(26,46,35,41)(27,42,29,43)(36,48,37,47);(10,14)(15,16)(17,20,23,21)(18,25,28,45)(19,31,34,32)(22,36,38,37)(24,40,30,44)(26,27,35,29)(33,47,39,48)(41,42,46,43);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,26,28,35)(19,32,34,31)(22,36,38,37)(24,41,30,46)(25,27,45,29)(33,47,39,48)(40,42,44,43);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,22,23,38)(18,27,28,29)(19,33,34,39)(20,36,21,37)(24,42,30,43)(25,35,45,26)(31,47,32,48)(40,46,44,41)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775553);