#8462123

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

4222 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=g,e^2=hi,f^2=ghi,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, 8369319);