#8114413

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

3991 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021609);