#9668371

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

4376 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=ghi,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575567);