#9815010

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

4139 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31)(33,34,38,40)(35,43,45,39)(36,41,44,48)(37,46,47,42);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,20)(18,24,22,25)(21,28,29,27)(26,32,30,31)(33,35,38,45)(34,39,40,43)(36,37,44,47)(41,42,48,46);(2,3)(7,9)(8,10)(14,15)(34,40)(39,43)(41,48)(42,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,25,22,24)(21,28,29,27)(26,32,30,31)(33,34)(35,39)(36,41)(37,42)(38,40)(43,45)(44,48)(46,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,36)(34,41)(35,37)(38,44)(39,42)(40,48)(43,46)(45,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,37)(34,42)(35,44)(36,45)(38,47)(39,48)(40,46)(41,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^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, 9722206);