#8746936

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

3819 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8654132);