#8096915

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

4196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=g,e^2=h,f^2=g,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, 8004111);