#8930313

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

4297 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,e^2=h,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, 8837509);