#9039155

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

3831 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,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, 8946351);