#9127581

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034777);