#8093461

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,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,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, 8000657);