#9814733

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(34,39)(40,42)(41,44)(47,48);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,14,16,11)(19,25)(23,26)(27,29)(31,32)(33,34)(35,40)(36,44)(37,42)(38,39)(41,45)(43,48)(46,47);(2,9)(3,4)(11,14)(12,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,35,38,37)(34,40,39,42)(36,43,45,46)(41,47,44,48);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(1,5)(2,11)(3,13)(4,12)(6,8)(7,15)(9,14)(10,16)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,36)(34,41)(35,43)(37,46)(38,45)(39,44)(40,47)(42,48);(1,6,7,16)(2,12,9,13)(3,14,4,11)(5,10,15,8)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,37,38,35)(34,42,39,40)(36,43,45,46)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=i,f^2=gh,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, 9721929);