#6495428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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

1363 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)(17,18,21,26,22,25,35,41)(19,27,30,37,29,42,34,23)(20,24,33,40,32,39,47,48)(28,43,44,38,31,45,46,36);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,25,37)(20,31,32,28)(21,34,35,30)(24,38,39,36)(26,42,41,27)(33,44,47,46)(40,43,48,45);(2,5)(6,8)(10,13)(14,16)(17,20,22,32)(18,24,25,39)(19,28,29,31)(21,33,35,47)(23,36,37,38)(26,40,41,48)(27,43,42,45)(30,44,34,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,29)(20,32)(23,37)(24,39)(27,42)(30,34)(33,47)(40,48);(9,11,12,15)(10,14,13,16)(18,25)(23,37)(24,39)(26,41)(27,42)(36,38)(40,48)(43,45)

Polycyclic presentation:

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

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, 6402624);