#8464229

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

4167 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=gh,e^2=i,f^2=gi,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, 8371425);