#8695849

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603045);