#9040905

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,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, 8948101);