#6367150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1612 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)(4,14)(5,11)(6,10)(7,12)(9,15)(13,16)(17,18,23,28)(19,29,33,38)(20,25,22,27)(21,26,36,44)(24,32,40,30)(31,45,47,42)(34,41,37,43)(35,48,46,39);(1,3,7,4)(2,8,12,14)(5,15,6,13)(9,10,16,11)(17,19)(18,24)(20,30)(21,35)(22,32)(23,33)(25,38)(26,42)(27,29)(28,40)(31,34)(36,46)(37,47)(39,43)(41,48)(44,45);(1,4,7,3)(2,9,12,16)(5,15,6,13)(8,10,14,11)(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,34,36,37)(24,38,40,29)(26,41,44,43)(31,46,47,35)(39,42,48,45);(17,21,23,36)(18,26,28,44)(19,31,33,47)(20,34,22,37)(24,39,40,48)(25,41,27,43)(29,45,38,42)(30,46,32,35);(1,5,7,6)(2,10,12,11)(3,13,4,15)(8,9,14,16)(17,20,23,22)(18,27,28,25)(19,32,33,30)(21,34,36,37)(24,38,40,29)(26,43,44,41)(31,35,47,46)(39,42,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274346);