#8930369

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

4117 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)(17,18,22,23)(19,24,28,32)(20,21,25,31)(26,30,29,27)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(18,23)(20,25)(24,32)(26,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(2,6)(3,9)(8,11)(10,12)(17,19,22,28)(18,24,23,32)(20,26,25,29)(21,30,31,27)(33,34,38,41)(35,39,43,36)(37,45,46,40)(42,44,48,47);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,32)(25,26)(30,31)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20,22,25)(18,21,23,31)(19,26,28,29)(24,30,32,27)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21)(18,25)(19,27)(20,23)(22,31)(24,26)(28,30)(29,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=gh,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, 8837565);