#8096801

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

4096 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003997);