#9947356

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

4115 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,22,19)(20,23,28,25)(21,26,30,24)(27,32,29,31)(33,34)(35,40)(36,39)(37,41)(38,44)(42,47)(43,45)(46,48);(3,9)(7,13)(10,15)(14,16)(18,19)(23,25)(24,26)(31,32)(33,35,39,41)(34,40,36,37)(38,43,47,48)(42,46,44,45);(2,6)(7,13)(8,11)(14,16)(18,19)(23,25)(24,26)(31,32)(34,36)(35,41)(42,44)(43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,36,39,34)(35,40,41,37)(38,44,47,42)(43,46,48,45);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37,39,40)(34,41,36,35)(38,46,47,45)(42,43,44,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 9854552);