#8926233

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833429);