#9815206

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

4112 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)(3,12)(4,5)(6,16)(7,9)(8,13)(10,15)(11,14)(17,18,22,27)(19,23,30,24)(20,25,31,29)(21,28,32,26)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,3,7,13)(2,8,9,12)(4,14,15,6)(5,16,10,11)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(35,41)(39,42)(43,45)(47,48);(2,9)(3,13)(4,15)(11,16)(17,19,22,30)(18,24,27,23)(20,28,31,26)(21,29,32,25)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,4)(2,10)(3,14)(5,9)(6,13)(7,15)(8,11)(12,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,5,7,10)(2,4,9,15)(3,11,13,16)(6,12,14,8)(17,20)(18,25)(19,28)(21,24)(22,31)(23,32)(26,30)(27,29)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,6,7,14)(2,11,9,16)(3,4,13,15)(5,12,10,8)(17,21,22,32)(18,26,27,28)(19,29,30,25)(20,23,31,24)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=hi,e^2=g,f^2=h,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, 9722402);