#8750710

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

4020 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 8657906);