#6912417

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#23923	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819613);