#8933089

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	#48091	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,e^2=g,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, 8840285);