#8720503

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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)(3,8)(4,10)(5,14)(6,12)(7,9)(11,15)(13,16)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,31)(30,32)(33,34)(35,38)(36,46)(37,48)(39,43)(40,45)(41,47)(42,44);(1,3)(2,8)(4,7)(5,13)(6,11)(9,10)(12,15)(14,16)(17,19,22,24)(18,21,25,26)(20,27,29,32)(23,31,28,30)(33,35)(34,40)(36,37)(38,43)(39,45)(41,48)(42,46)(44,47);(2,9)(3,4)(11,14)(12,13)(20,29)(23,28)(27,32)(30,31)(33,36,39,47)(34,41,43,46)(35,37,45,44)(38,42,40,48);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,20)(18,23)(19,27)(21,31)(22,29)(24,32)(25,28)(26,30)(33,36)(34,41)(35,44)(37,45)(38,48)(39,47)(40,42)(43,46);(1,5,7,15)(2,11,9,14)(3,13,4,12)(6,10,16,8)(17,18)(19,21)(20,23)(22,25)(24,26)(27,31)(28,29)(30,32)(33,37,39,44)(34,42,43,48)(35,36,45,47)(38,41,40,46);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,8,15,10)(17,21)(18,24)(19,25)(20,30)(22,26)(23,27)(28,32)(29,31)(33,38,39,40)(34,35,43,45)(36,42,47,48)(37,46,44,41)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627699);