#6434106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1730 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)(17,18,22,27)(19,28,31,41)(20,33,36,42)(21,25,37,26)(23,30,39,32)(24,34,40,35)(29,38,45,47)(43,46,48,44);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,23,27,39)(20,29,36,45)(21,30,37,32)(24,38,40,47)(25,28,26,41)(33,43,42,48)(34,44,35,46);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,36)(18,24,27,40)(19,29,31,45)(21,34,37,35)(23,38,39,47)(25,33,26,42)(28,43,41,48)(30,44,32,46);(10,14)(13,16)(17,21,22,37)(18,25,27,26)(19,30,31,32)(20,34,36,35)(23,28,39,41)(24,33,40,42)(29,44,45,46)(38,43,47,48);(19,31)(23,39)(28,41)(29,45)(30,32)(38,47)(43,48)(44,46)

Polycyclic presentation:

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

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, 6341302);