#6394061

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342866
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2727 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,23,28,22,27,21,26)(19,29,31,41,32,24,36,39)(20,33,34,42,35,25,37,40)(30,38,46,48,45,43,44,47);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,32)(18,24,27,29)(20,30,35,45)(21,36,23,31)(25,38,33,43)(26,41,28,39)(34,46,37,44)(40,48,42,47);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,23,37,22,35,21,34)(18,25,26,40,27,33,28,42)(19,30,31,44,32,45,36,46)(24,38,41,48,29,43,39,47);(3,12)(7,10)(11,13)(15,16)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,40,33,42)(30,44,45,46)(38,47,43,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,27)(19,32)(25,33)(26,28)(30,45)(31,36)(40,42)(44,46)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301257);