#8461192

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

4207 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,e^2=hi,f^2=i,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, 8368388);