#6468654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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

1508 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)(10,14)(15,16)(17,18,21,26,22,25,36,42)(19,27,30,38,29,43,35,23)(20,31,34,47,33,24,39,41)(28,45,46,37,32,44,48,40);(9,10,13,14)(11,16,12,15)(17,19,22,29)(18,23,25,38)(20,32,33,28)(21,35,36,30)(24,40,31,37)(26,43,42,27)(34,46,39,48)(41,44,47,45);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,22,33)(18,24,25,31)(19,28,29,32)(21,34,36,39)(23,37,38,40)(26,41,42,47)(27,44,43,45)(30,46,35,48);(9,12,13,11)(10,16,14,15)(18,25)(19,29)(24,31)(26,42)(28,32)(30,35)(41,47)(46,48);(1,3,4,7)(2,6,5,8)(20,33)(24,31)(28,32)(34,39)(37,40)(41,47)(44,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375850);