#8461191

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

4211 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)(19,27)(20,29)(26,31)(33,34)(35,43)(36,40)(37,45)(38,41)(39,42)(44,47)(46,48);(1,2,6,5)(3,8,11,7)(4,9,12,13)(10,15,16,14)(17,18,23,24)(19,22,27,32)(20,30,29,25)(21,26,28,31)(33,35)(34,37)(36,44)(38,40)(39,45)(41,46)(42,43)(47,48);(1,3)(2,7)(4,10)(5,8)(6,11)(9,14)(12,16)(13,15)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,26,30,31)(33,36,39,46)(34,40,42,48)(35,44,45,41)(37,38,43,47);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,20)(18,25)(19,28)(21,27)(22,26)(23,29)(24,30)(31,32)(33,37)(34,35)(36,38)(39,43)(40,44)(41,48)(42,45)(46,47);(1,4,6,12)(2,9,5,13)(3,10,11,16)(7,14,8,15)(17,21,23,28)(18,26,24,31)(19,20,27,29)(22,30,32,25)(33,35)(34,37)(36,44)(38,40)(39,45)(41,46)(42,43)(47,48);(1,5,6,2)(3,8,11,7)(4,13,12,9)(10,15,16,14)(17,22)(18,27)(19,24)(20,31)(21,30)(23,32)(25,28)(26,29)(33,38)(34,41)(35,40)(36,43)(37,46)(39,47)(42,44)(45,48)

Polycyclic presentation:

2^9|b^2=hi,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, 8368387);