#9815227

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

4278 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)(3,9)(4,13)(5,14)(6,8)(7,12)(10,11)(15,16)(17,18,23,25)(19,28,29,24)(20,30,27,22)(21,26,31,32)(33,34,38,43)(35,39,46,40)(36,41,47,45)(37,44,48,42);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,15)(14,16)(17,19)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(2,8)(4,14)(7,9)(10,16)(18,25)(19,29)(20,27)(26,32)(33,35,38,46)(34,40,43,39)(36,44,47,42)(37,45,48,41);(1,4)(2,5)(3,10)(6,14)(7,11)(8,13)(9,15)(12,16)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(1,3)(2,9)(4,10)(5,15)(6,12)(7,8)(11,13)(14,16)(17,21)(18,26)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(33,36)(34,41)(35,44)(37,40)(38,47)(39,48)(42,46)(43,45);(1,5)(2,4)(3,11)(6,13)(7,10)(8,14)(9,16)(12,15)(17,22)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,37,38,48)(34,42,43,44)(35,45,46,41)(36,39,47,40)

Polycyclic presentation:

2^9|a^2=i,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, 9722423);