#6769391

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1284 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,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,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, 6676587);