#9812878

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	#46792	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4223 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:

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

Polycyclic presentation:

2^9|a^2=ghi,d^2=gi,f^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, 9720074);