#8926070

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

4072 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833266);