#8726485

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

3873 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,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, 8633681);