#10504998

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

4088 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)(4,14)(5,9)(6,15)(7,11)(10,12)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3,7,13)(2,8,11,16)(4,12,5,6)(9,15,14,10)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,4,7,5)(2,9,11,14)(3,12,13,6)(8,15,16,10)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,5,7,4)(2,9,11,14)(3,6,13,12)(8,15,16,10)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,6)(2,10)(3,4)(5,13)(7,12)(8,9)(11,15)(14,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10412194);