#10246858

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	#48309	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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)(3,10,9,5)(4,8,12,16)(11,14,13,15)(17,18,23,27)(19,28,21,24)(20,29,31,25)(22,26,32,30)(33,34,38,39)(35,41,42,36)(37,45,46,40)(43,44,48,47);(2,7)(3,9)(4,12)(14,15)(17,19,23,21)(18,24,27,28)(20,30,31,26)(22,29,32,25)(33,35,38,42)(34,36,39,41)(37,43,46,48)(40,44,45,47);(2,7)(3,9)(8,16)(11,13)(34,39)(35,42)(40,45)(43,48);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,14,16,15)(17,20,23,31)(18,25,27,29)(19,26,21,30)(22,24,32,28)(33,34)(35,36)(37,40)(38,39)(41,42)(43,44)(45,46)(47,48);(1,4)(2,8)(3,11)(5,15)(6,12)(7,16)(9,13)(10,14)(17,21,23,19)(18,24,27,28)(20,26,31,30)(22,29,32,25)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,5)(2,9)(3,7)(4,14)(6,10)(8,11)(12,15)(13,16)(17,22)(18,26)(19,29)(20,24)(21,25)(23,32)(27,30)(28,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,47)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154054);