#8926638

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4824 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833834);