#8930368

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

4115 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)(19,24)(20,25)(21,26)(22,23)(27,32)(28,29)(30,31)(33,34,39,37)(35,42,40,36)(38,45,47,41)(43,44,48,46);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(18,23)(20,24)(26,31)(28,32)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(2,6)(3,9)(8,11)(10,12)(17,19,22,25)(18,24,23,20)(21,29,30,27)(26,28,31,32)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,32)(28,31)(29,30)(33,36)(34,35)(37,40)(38,44)(39,42)(41,43)(45,48)(46,47);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20,22,24)(18,25,23,19)(21,28,30,32)(26,29,31,27)(33,37,39,34)(35,42,40,36)(38,45,47,41)(43,46,48,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,32)(25,29)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837564);