#6874854

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 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)(17,18,21,27,22,23,37,26)(19,28,32,38,29,42,31,24)(20,33,36,41,35,25,34,40)(30,45,46,39,44,43,47,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(18,23)(19,29)(25,33)(26,27)(30,44)(31,32)(40,41)(46,47);(3,8)(5,7)(10,14)(15,16)(17,19,22,29)(18,24,23,38)(20,30,35,44)(21,31,37,32)(25,39,33,48)(26,28,27,42)(34,46,36,47)(40,45,41,43);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,33)(24,39)(26,41)(27,40)(28,43)(29,44)(31,47)(32,46)(34,37)(38,48)(42,45);(9,12,13,11)(10,15,14,16)(17,21,22,37)(18,26,23,27)(19,31,29,32)(20,34,35,36)(24,28,38,42)(25,40,33,41)(30,46,44,47)(39,45,48,43)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,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, 6782050);