#6546528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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, 6453724);