#6434093

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1753 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fg,e^2=i,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,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 6341289);