#9863298

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

4110 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)(3,8)(4,5)(6,15)(7,9)(10,12)(11,16)(13,14)(17,18)(19,21)(20,28)(22,26)(23,25)(24,30)(27,31)(29,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,3,7,11)(2,8,9,16)(4,12,13,6)(5,10,14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32)(34,40)(35,37)(42,46)(43,45);(2,9)(3,11)(4,13)(10,15)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,29,23,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,4)(2,5)(3,12)(6,11)(7,13)(8,10)(9,14)(15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,5)(2,4)(3,10)(6,16)(7,14)(8,12)(9,13)(11,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,31)(30,32)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,6,7,12)(2,10,9,15)(3,4,11,13)(5,8,14,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,28)(30,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=ghi,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, 9770494);