#10349030

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

4300 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)(3,12)(4,8)(5,9)(6,13)(7,11)(10,16)(14,15)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(3,4)(6,16)(8,12)(10,13)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,3,7,4)(2,8,11,12)(5,10,15,13)(6,14,16,9)(17,20,22,28)(18,23,25,19)(21,27,30,31)(24,29,32,26)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4,7,3)(2,8,11,12)(5,10,15,13)(6,9,16,14)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,5,7,15)(2,9,11,14)(3,13,4,10)(6,8,16,12)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,6,7,16)(2,10,11,13)(3,14,4,9)(5,8,15,12)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=h,e^2=gh,f^2=h,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, 10256226);