#6436060

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1505 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)(11,15)(14,16)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,32)(28,40)(29,38)(31,46)(33,47)(34,41)(35,48)(36,43)(39,45)(42,44);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,34,36,29)(25,30,40,47)(26,41,43,38)(31,45,35,44)(39,48,42,46);(10,13)(14,16)(17,20,22,32,23,33,19,28)(18,25,24,30,27,40,37,47)(21,35,34,44,36,31,29,45)(26,42,38,48,43,39,41,46);(9,11,12,15)(10,14,13,16)(17,21,23,36)(18,26,27,43)(19,29,22,34)(20,31,33,35)(24,38,37,41)(25,39,40,42)(28,44,32,45)(30,46,47,48);(1,3,4,7)(2,6,5,8)(21,36)(26,43)(29,34)(31,35)(38,41)(39,42)(44,45)(46,48)

Polycyclic presentation:

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

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, 6343256);