#8726700

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gi,e^2=i,f^2=gi,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, 8633896);