#8434122

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

4288 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,e^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, 8341318);