#8462112

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

4109 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=g,e^2=ghi,f^2=hi,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, 8369308);