#8492821

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

4257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8400017);