#9040397

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(33,34,40,44)(35,45,47,41)(36,38,37,42)(39,48,46,43);(2,3)(4,11)(8,10)(12,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35,40,47)(34,41,44,45)(36,43,37,48)(38,46,42,39);(4,11)(5,13)(7,9)(8,10)(18,22)(20,29)(23,25)(26,30)(33,36,40,37)(34,42,44,38)(35,43,47,48)(39,45,46,41);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,37,40,36)(34,38,44,42)(35,43,47,48)(39,45,46,41);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,38)(34,37)(35,46)(36,44)(39,47)(40,42)(41,48)(43,45);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(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=h,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, 8947593);