#6560026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6467222);