#8463623

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

4037 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^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, 8370819);