#8479514

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

4091 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)(3,9,4,8)(5,14,15,10)(6,11,16,13)(17,18)(19,23)(20,29)(21,24)(22,25)(26,32)(27,30)(28,31)(33,34,39,42)(35,43,46,40)(36,47,48,37)(38,41,45,44);(1,3,7,4)(2,8,12,9)(5,11,15,13)(6,10,16,14)(17,19,22,28)(18,23,25,31)(20,27,21,26)(24,32,29,30)(33,35)(34,40)(36,44)(37,38)(39,46)(41,48)(42,43)(45,47);(17,20,22,21)(18,24,25,29)(19,26,28,27)(23,30,31,32)(33,36)(34,37)(35,44)(38,40)(39,48)(41,46)(42,47)(43,45);(1,4,7,3)(2,9,12,8)(5,11,15,13)(6,10,16,14)(17,21,22,20)(18,24,25,29)(19,27,28,26)(23,30,31,32)(33,36)(34,37)(35,41)(38,43)(39,48)(40,45)(42,47)(44,46);(1,5)(2,10)(3,13)(4,11)(6,8)(7,15)(9,16)(12,14)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37)(34,36)(35,45)(38,46)(39,47)(40,41)(42,48)(43,44);(1,6,7,16)(2,11,12,13)(3,10,4,14)(5,8,15,9)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,32)(28,31)(33,38,39,45)(34,41,42,44)(35,37,46,47)(36,40,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=gh,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, 8386710);