#8930297

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

4125 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=hi,e^2=gh,f^2=g,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, 8837493);