#8957603

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

4208 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8864799);