#8097773

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

4132 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)(7,14)(10,16)(13,15)(17,18)(19,21)(20,29)(22,25)(23,26)(24,28)(27,32)(30,31)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,47)(45,48);(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,26)(22,27)(23,28)(25,31)(29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,23,22)(18,25,26,29)(19,27,28,30)(21,31,24,32)(33,36,38,45)(34,40,42,48)(35,44,46,37)(39,41,43,47);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,21)(18,19)(20,31)(22,32)(23,24)(25,27)(26,28)(29,30);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,19)(18,21)(20,27)(22,30)(23,28)(24,26)(25,31)(29,32)(33,37,38,44)(34,41,42,47)(35,45,46,36)(39,40,43,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,22,23,20)(18,25,26,29)(19,27,28,30)(21,32,24,31)(33,34)(35,39)(36,40)(37,47)(38,42)(41,44)(43,46)(45,48)

Polycyclic presentation:

2^9|c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

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, 8004969);