#10183120

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	#43255	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4006 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)(17,18,22,25)(19,26,20,23)(21,24,29,32)(27,31,28,30)(33,34,40,44)(35,41,38,43)(36,42,37,39)(45,46,47,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,32,31)(33,35,40,38)(34,41,44,43)(36,47,37,45)(39,46,42,48);(2,7)(3,10)(8,15)(11,12)(33,36,40,37)(34,42,44,39)(35,45,38,47)(41,46,43,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,31)(29,32)(33,37,40,36)(34,39,44,42)(35,45,38,47)(41,46,43,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20,22,19)(18,23,25,26)(21,27,29,28)(24,31,32,30)(33,38,40,35)(34,43,44,41)(36,47,37,45)(39,46,42,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,32)(26,31)(33,39)(34,36)(35,46)(37,44)(38,48)(40,42)(41,47)(43,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10090316);