#8994400

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

4163 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(18,23)(19,25)(26,30)(27,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(1,3,7,4)(2,8,9,10)(5,14,15,12)(6,13,16,11)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(34,38)(39,41)(40,44)(47,48);(2,9)(3,4)(5,15)(6,16)(19,25)(20,24)(21,29)(26,30)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=i,e^2=gh,f^2=g,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, 8901596);