#6350947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2851 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)(3,13)(4,9)(5,11)(6,10)(7,12)(8,14)(15,16)(17,18,22,26)(19,27,29,40)(20,24,31,38)(21,33,34,25)(23,35,37,30)(28,41,42,48)(32,45,44,39)(36,46,47,43);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(21,34)(25,33)(27,40)(30,35)(32,44)(39,45)(41,48)(43,46);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,19)(18,23)(20,28)(21,35)(22,29)(24,36)(25,40)(26,37)(27,33)(30,34)(31,42)(32,46)(38,47)(39,48)(41,45)(43,44);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(17,20)(18,24)(19,28)(21,32)(22,31)(23,36)(25,39)(26,38)(27,41)(29,42)(30,43)(33,45)(34,44)(35,46)(37,47)(40,48);(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258143);