#6469915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1407 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)(4,8)(5,6)(17,18,22,26,23,27,37,42)(19,28,31,39,32,41,36,24)(20,25,33,40,21,34,35,48)(29,43,44,47,30,45,46,38);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,32)(18,24,27,39)(20,29,21,30)(22,36,37,31)(25,38,34,47)(26,41,42,28)(33,46,35,44)(40,45,48,43);(2,6)(7,8)(10,13)(14,16)(17,20,23,21)(18,25,27,34)(19,29,32,30)(22,33,37,35)(24,38,39,47)(26,40,42,48)(28,43,41,45)(31,44,36,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,21,23,20)(18,25,27,34)(19,30,32,29)(22,35,37,33)(24,38,39,47)(26,40,42,48)(28,43,41,45)(31,46,36,44);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(20,21)(25,34)(29,30)(33,35)(38,47)(40,48)(43,45)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6377111);