#9167576

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,21,25,23)(20,24,28,32)(27,30,29,31)(33,34,39,40)(35,43,44,36)(37,41,47,48)(38,46,45,42);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32)(33,35,39,44)(34,36,40,43)(37,45,47,38)(41,46,48,42);(3,10)(6,7)(11,16)(14,15)(20,28)(24,32)(27,29)(30,31)(34,40)(36,43)(41,48)(42,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,28)(18,24,26,32)(19,27,25,29)(21,30,23,31)(33,36,39,43)(34,35,40,44)(37,46,47,42)(38,41,45,48);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,19)(18,21)(20,29)(22,25)(23,26)(24,31)(27,28)(30,32)(33,37,39,47)(34,41,40,48)(35,38,44,45)(36,42,43,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,25)(19,26)(20,30)(22,23)(24,29)(27,32)(28,31)(33,38)(34,42)(35,37)(36,41)(39,45)(40,46)(43,48)(44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9074772);