#6355466

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355448
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1540 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)(3,7)(4,9)(5,14)(6,10)(8,11)(12,15)(13,16)(17,18,23,27)(19,28,32,38)(20,25,35,42)(21,26,22,36)(24,31,39,30)(29,43,46,48)(33,40,34,41)(37,44,47,45);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,16)(10,15)(17,19,21,31,23,32,22,30)(18,24,36,38,27,39,26,28)(20,29,34,44,35,46,33,45)(25,37,40,48,42,47,41,43);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,35)(24,37)(26,41)(27,42)(28,43)(30,45)(31,44)(32,46)(36,40)(38,48)(39,47);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,15,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,38,39,28)(25,40,42,41)(29,44,46,45)(37,43,47,48);(2,10)(7,15)(8,16)(9,14)(17,21,23,22)(18,26,27,36)(19,31,32,30)(20,34,35,33)(24,28,39,38)(25,41,42,40)(29,44,46,45)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=dhi,d^c=dh,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, 6262662);