#9041002

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	#35438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,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, 8948198);