#10349045

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

4094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=h,e^2=h,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, 10256241);