#6909669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#23962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1799 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,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, 6816865);