#8726411

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

3871 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)(19,24)(20,29)(21,26)(22,23)(25,30)(27,28)(31,32)(33,34)(35,44)(36,46)(37,45)(38,41)(39,40)(42,47)(43,48);(1,3,7,10)(2,8,5,4)(6,11,15,16)(9,14,13,12)(17,18)(19,27)(20,25)(21,26)(22,23)(24,28)(29,30)(31,32)(34,40)(36,47)(41,44)(43,45);(3,10)(4,8)(11,16)(12,14)(18,23)(19,28)(25,29)(26,32)(33,35,39,38)(34,41,40,44)(36,43,47,45)(37,42,48,46);(1,4)(2,3)(5,10)(6,14)(7,8)(9,16)(11,13)(12,15)(17,19)(18,24)(20,26)(21,29)(22,28)(23,27)(25,31)(30,32)(33,36,39,47)(34,42,40,46)(35,45,38,43)(37,44,48,41);(1,5,7,2)(3,4,10,8)(6,9,15,13)(11,14,16,12)(17,20)(18,25)(19,26)(21,27)(22,30)(23,29)(24,31)(28,32)(33,37)(34,43)(35,42)(36,44)(38,46)(39,48)(40,45)(41,47);(1,6,7,15)(2,9,5,13)(3,11,10,16)(4,12,8,14)(17,21,22,31)(18,26,23,32)(19,25,28,29)(20,24,30,27)(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=ghi,c^2=g,d^2=g,e^2=ghi,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, 8633607);