#8093456

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

4003 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)(10,16)(14,15)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,30)(31,32)(33,34)(35,43)(36,46)(37,41)(38,42)(39,40)(44,48)(45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31)(34,40)(36,43)(41,42)(47,48);(2,6)(7,9)(8,13)(14,15)(19,26)(22,25)(27,32)(30,31)(33,35)(34,36)(37,44)(38,45)(39,46)(40,43)(41,47)(42,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,36)(34,35)(37,47)(38,48)(39,43)(40,46)(41,44)(42,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,37,39,38)(34,41,40,42)(35,44,46,45)(36,47,43,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,38,39,37)(34,42,40,41)(35,45,46,44)(36,48,43,47)

Polycyclic presentation:

2^9|b^2=gh,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,d^c=dghi

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, 8000652);