#6546611

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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)(17,18,22,26)(19,27,29,43)(20,31,33,24)(21,35,36,25)(23,37,39,30)(28,44,32,42)(34,41,45,47)(38,46,40,48);(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,29,36,37)(18,23,25,27,26,39,35,43)(20,32,34,48,33,28,45,46)(24,40,41,44,31,38,47,42);(2,5)(6,8)(10,13)(14,16)(17,20,22,33)(18,24,26,31)(19,28,29,32)(21,34,36,45)(23,38,39,40)(25,41,35,47)(27,42,43,44)(30,46,37,48);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,29)(20,33)(23,39)(24,31)(27,43)(30,37)(34,45)(41,47);(11,15)(14,16)(19,29)(23,39)(27,43)(28,32)(30,37)(38,40)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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, 6453807);