#6557965

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fi,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465161);