#8750707

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3958 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)(3,11)(4,10)(5,9)(6,12)(7,8)(13,16)(14,15)(17,18,23,26)(19,21,28,25)(20,29,22,24)(27,31,30,32)(34,38)(36,43)(39,44)(42,47);(2,8)(4,14)(9,11)(12,13)(17,19)(18,21)(20,30)(22,27)(23,28)(24,31)(25,26)(29,32)(35,41)(39,44)(42,47)(46,48);(1,3,7,5)(2,9,8,11)(4,12,14,13)(6,10,16,15)(17,20,23,22)(18,24,26,29)(19,27,28,30)(21,32,25,31)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,7,14)(2,10,8,15)(3,13,5,12)(6,11,16,9)(17,21,23,25)(18,19,26,28)(20,31,22,32)(24,30,29,27)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5,7,3)(2,11,8,9)(4,12,14,13)(6,10,16,15)(17,19)(18,25)(20,27)(21,26)(22,30)(23,28)(24,31)(29,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48);(1,6,7,16)(2,12,8,13)(3,10,5,15)(4,9,14,11)(17,22,23,20)(18,24,26,29)(19,27,28,30)(21,31,25,32)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gi,e^2=g,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, 8657903);