#10086329

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	#52328	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(34,36)(35,38)(41,44)(43,45);(2,6)(3,9)(8,11)(10,12)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(2,6)(3,9)(8,11)(10,12)(19,25)(22,29)(26,27)(30,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,19,21,25)(18,22,24,29)(20,27,28,26)(23,31,32,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9993525);