#8418183

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

4287 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8325379);