#6350958

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

2769 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 6258154);