#9947387

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4111 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9854583);