#8700447

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

3863 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607643);