#10349004

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

4312 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10256200);