#6434564

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1342 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,33,39)(20,34,37,41)(21,26,22,27)(24,31,40,32)(25,35,42,36)(30,38,46,48)(43,45,47,44);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,33,21,32)(18,24,26,39,28,40,27,29)(20,30,36,44,37,46,35,45)(25,38,34,47,42,48,41,43);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,21,35,23,37,22,36)(18,25,27,41,28,42,26,34)(19,30,32,45,33,46,31,44)(24,38,29,43,40,48,39,47);(10,14)(13,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,40,39)(25,34,42,41)(30,44,46,45)(38,43,48,47);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

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

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, 6341760);