#8934302

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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)(3,7)(4,6)(5,8)(9,11)(10,14)(12,13)(15,16)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25)(33,34,40,44)(35,41,46,48)(36,37,38,42)(39,45,47,43);(2,4)(3,9)(8,12)(10,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,35,40,46)(34,41,44,48)(36,47,38,39)(37,43,42,45);(3,9)(5,13)(7,11)(8,12)(19,27)(20,30)(23,31)(24,29)(33,36,40,38)(34,42,44,37)(35,39,46,47)(41,43,48,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37,40,42)(34,36,44,38)(35,45,46,43)(39,48,47,41);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29)(33,38,40,36)(34,37,44,42)(35,39,46,47)(41,43,48,45);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)(33,39)(34,43)(35,36)(37,48)(38,46)(40,47)(41,42)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841498);