#9873188

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3812 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(17,18,23,22)(19,25,20,24)(21,26,29,30)(27,32,28,31);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,19,23,20)(18,24,22,25)(21,28,29,27)(26,32,30,31)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(2,6)(7,14)(8,11)(15,16)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,20,23,19)(18,25,22,24)(21,28,29,27)(26,32,30,31)(33,36,39,34)(35,41,37,40)(38,44,46,42)(43,48,45,47);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,21,23,29)(18,26,22,30)(19,27,20,28)(24,31,25,32)(33,37,39,35)(34,41,36,40)(38,43,46,45)(42,47,44,48);(1,4)(2,8)(3,9)(5,13)(6,11)(7,15)(10,12)(14,16)(17,22,23,18)(19,25,20,24)(21,26,29,30)(27,31,28,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9780384);