#9815228

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

4280 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,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, 9722424);