#9668302

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575498);