#9712238

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

4098 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)(3,11)(4,13)(5,10)(6,14)(7,9)(8,12)(15,16)(17,18,22,23)(19,27,30,24)(20,25,26,21)(28,29,32,31)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3,7,12)(2,8,9,11)(4,5,14,15)(6,10,13,16)(18,23)(20,26)(24,27)(28,32)(35,37)(40,42)(43,44)(47,48);(2,9)(3,12)(4,14)(10,16)(17,19,22,30)(18,24,23,27)(20,28,26,32)(21,29,25,31)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4)(2,6)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,27)(25,31)(26,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,5)(2,10)(3,4)(6,8)(7,15)(9,16)(11,13)(12,14)(17,20)(18,25)(19,28)(21,23)(22,26)(24,31)(27,29)(30,32)(33,37,39,35)(34,40,38,42)(36,44,46,43)(41,47,45,48);(1,6)(2,4)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,21,22,25)(18,26,23,20)(19,29,30,31)(24,32,27,28)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9619434);