#9175024

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	#53989	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=gh,f^2=g,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, 9082220);