#9783070

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	#53937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,42)(36,44)(37,40)(38,41)(39,46)(43,45)(47,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35,38,40)(34,37,41,42)(36,43,46,47)(39,45,44,48);(2,7)(3,9)(8,14)(10,15)(18,25)(19,20)(26,32)(27,28);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36)(34,39)(35,43)(37,45)(38,46)(40,47)(41,44)(42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31)(33,37,38,42)(34,40,41,35)(36,45,46,48)(39,47,44,43);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^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, 9690266);