#8726501

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

3881 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=g,e^2=hi,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, 8633697);