#6629975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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

1769 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,27,36)(20,30,32,40)(21,33,35,25)(24,34,39,31)(28,43,41,48)(29,45,42,38)(37,46,47,44);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,37)(25,38)(26,36)(30,43)(31,44)(32,41)(33,45)(34,46)(35,42)(39,47)(40,48);(10,14)(13,16)(19,27)(23,36)(28,41)(29,42)(37,47)(38,45)(43,48)(44,46);(17,20,22,32)(18,24,26,39)(19,28,27,41)(21,34,35,31)(23,37,36,47)(25,40,33,30)(29,46,42,44)(38,48,45,43);(1,3,4,7)(2,6,5,8)(19,27)(23,36)(28,41)(29,42)(37,47)(38,45)(43,48)(44,46)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537171);