#9814867

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	#58172	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=gi,e^2=g,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, 9722063);