#8726359

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3894 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=h,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, 8633555);