#10328089

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

4140 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 10235285);