#10247606

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

4133 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:

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

Polycyclic presentation:

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

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, 10154802);