#10504604

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

4287 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)(3,7)(4,8)(5,11)(6,9)(10,13)(12,15)(14,16)(17,18)(19,26)(20,29)(21,24)(22,28)(23,25)(27,31)(30,32)(33,34,38,43)(35,42,45,39)(36,46,47,40)(37,41,44,48);(1,3,6,11)(2,7,9,5)(4,10,14,12)(8,13,16,15)(17,19)(18,22)(20,30)(21,27)(23,28)(24,32)(25,26)(29,31)(33,35)(34,39)(36,44)(37,47)(38,45)(40,48)(41,46)(42,43);(1,3)(2,5)(4,10)(6,11)(7,9)(8,15)(12,14)(13,16)(17,20,23,21)(18,24,25,29)(19,27,28,30)(22,31,26,32)(33,36,38,47)(34,40,43,46)(35,44,45,37)(39,48,42,41);(1,3)(2,5)(4,12)(6,11)(7,9)(8,13)(10,14)(15,16)(17,19)(18,22)(20,27)(21,30)(23,28)(24,31)(25,26)(29,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,46)(43,48)(45,47);(1,4,6,14)(2,8,9,16)(3,10,11,12)(5,15,7,13)(17,21,23,20)(18,24,25,29)(19,27,28,30)(22,32,26,31)(33,35)(34,42)(36,44)(37,47)(38,45)(39,43)(40,41)(46,48);(1,5)(2,3)(4,13)(6,7)(8,12)(9,11)(10,16)(14,15)(17,22)(18,19)(20,31)(21,32)(23,26)(24,27)(25,28)(29,30)(33,34)(35,39)(36,40)(37,41)(38,43)(42,45)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=i,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, 10411800);