#6630907

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,24,30)(19,27,33,39)(20,26,35,42)(21,25,32,40)(22,29,23,28)(31,41,45,48)(34,38,46,47)(36,44,37,43);(2,5)(6,8)(17,19,22,21,24,33,23,32)(18,25,28,27,30,40,29,39)(20,31,36,34,35,45,37,46)(26,38,43,41,42,47,44,48);(10,13)(14,16)(17,20,24,35)(18,26,30,42)(19,31,33,45)(21,34,32,46)(22,36,23,37)(25,38,40,47)(27,41,39,48)(28,43,29,44);(1,3,4,7)(2,6,5,8)(17,21,22,33,24,32,23,19)(18,27,28,40,30,39,29,25)(20,34,36,45,35,46,37,31)(26,41,43,47,42,48,44,38);(9,11,12,15)(10,14,13,16)(20,35)(26,42)(31,45)(34,46)(36,37)(38,47)(41,48)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,d^b=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538103);