#8505488

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

4100 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gi,e^2=g,f^2=hi,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, 8412684);