#6567437

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

1314 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,16,12,15)(17,18,23,28)(19,29,31,40)(20,34,35,42)(21,26,22,27)(24,33,39,32)(25,37,41,36)(30,38,46,48)(43,45,47,44);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,19,22,33,23,31,21,32)(18,24,26,40,28,39,27,29)(20,30,36,44,35,46,37,45)(25,38,34,47,41,48,42,43);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,20,21,37,23,35,22,36)(18,25,27,42,28,41,26,34)(19,30,32,45,31,46,33,44)(24,38,29,43,39,48,40,47);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(19,31)(20,35)(24,39)(25,41)(29,40)(32,33)(34,42)(36,37);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,36,35,37)(24,40,39,29)(25,42,41,34)(30,44,46,45)(38,43,48,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=f,d^2=fg,e^2=fgi,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, 6474633);