#8933200

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4106 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,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, 8840396);