#8093213

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

4130 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)(19,27)(20,23)(21,25)(22,32)(24,28)(26,29)(30,31)(36,43)(40,45)(42,47)(46,48);(3,9)(4,11)(7,12)(8,13)(18,20)(19,28)(21,30)(26,32)(34,38)(35,41)(40,45)(42,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,19)(18,24)(20,27)(21,29)(22,30)(23,28)(25,32)(26,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,23,18)(19,27,28,24)(21,31,30,25)(22,32,29,26);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21,23,30)(18,25,20,31)(19,29,28,22)(24,32,27,26)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,22,23,29)(18,26,20,32)(19,30,28,21)(24,31,27,25)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000409);