#9716135

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	#45914	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9623331);