#10069431

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(19,25)(23,26)(27,31)(30,32)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(1,2,5,8)(3,6,9,13)(4,7,12,16)(10,14,11,15)(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30)(36,44)(40,47)(42,48)(45,46);(3,9)(6,13)(10,11)(14,15)(18,20)(23,26)(24,28)(30,32)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,36,38,44)(34,40,41,47)(35,42,43,48)(37,45,39,46);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=i,e^2=hi,f^2=g,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, 9976627);