#9814990

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3819 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=hi,e^2=g,f^2=h,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, 9722186);