#6554112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^c=ei,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, 6461308);