#8932534

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	#35676	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5234 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839730);