#9815236

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,24)(19,20,27,23)(21,25,30,32)(26,28,29,31)(33,34,38,43)(35,39,46,40)(36,41,47,45)(37,44,48,42);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,32)(29,30)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(2,7)(3,10)(8,15)(11,12)(18,24)(20,23)(25,32)(28,31)(33,35,38,46)(34,40,43,39)(36,44,47,42)(37,45,48,41);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,32,29)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21,22,30)(18,25,24,32)(19,26,27,29)(20,28,23,31)(33,36)(34,41)(35,44)(37,40)(38,47)(39,48)(42,46)(43,45);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,37,38,48)(34,42,43,44)(35,45,46,41)(36,39,47,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=hi,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, 9722432);