#8496876

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

4113 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404072);