#8097785

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

4146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=gh,f^2=g,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, 8004981);