#8994002

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

4229 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8901198);