#9040319

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	#50711	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947515);