#8434154

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

4232 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8341350);