#10166424

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gi,e^2=gi,f^2=g,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, 10073620);