#9863441

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

4223 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,24)(19,25,29,23)(20,30,26,21)(27,28,32,31)(33,34)(35,40)(36,37)(38,47)(39,43)(41,46)(42,44)(45,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,25)(26,32)(30,31)(33,35,39,45)(34,40,43,48)(36,44,46,38)(37,42,41,47);(2,7)(3,10)(8,15)(11,12)(18,24)(20,26)(23,25)(27,32)(33,36,39,46)(34,41,43,37)(35,38,45,44)(40,42,48,47);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,22,26)(18,21,24,30)(19,27,29,32)(23,28,25,31)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,19)(18,25)(20,27)(21,31)(22,29)(23,24)(26,32)(28,30)(33,37)(34,36)(35,42)(38,48)(39,41)(40,44)(43,46)(45,47);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,26)(19,28)(20,24)(22,30)(23,32)(25,27)(29,31)(33,38,39,44)(34,42,43,47)(35,36,45,46)(37,40,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

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, 9770637);