#6784763

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1519 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)(17,18,23,27)(19,28,31,38)(20,25,32,40)(21,26,22,36)(24,34,39,30)(29,43,44,48)(33,41,35,42)(37,45,47,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(20,32)(25,40)(29,44)(33,35)(37,47)(41,42)(43,48)(45,46);(2,6)(5,7)(10,13)(14,16)(17,19,21,34,23,31,22,30)(18,24,36,38,27,39,26,28)(20,29,35,45,32,44,33,46)(25,37,41,48,40,47,42,43);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,23,32)(18,25,27,40)(19,29,31,44)(21,35,22,33)(24,37,39,47)(26,42,36,41)(28,43,38,48)(30,46,34,45);(3,8)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,30,31,34)(20,33,32,35)(24,38,39,28)(25,41,40,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6691959);