#8096921

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

4184 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)(3,9,4,6)(5,10,14,16)(11,13,12,15)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34,39,37)(35,42,40,36)(38,41,46,45)(43,48,47,44);(2,8)(3,4)(10,16)(11,12)(20,28)(24,31)(27,32)(29,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,3,7,4)(2,6,8,9)(5,11,14,12)(10,15,16,13)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,34,39,37)(35,36,40,42)(38,41,46,45)(43,44,47,48);(1,4,7,3)(2,9,8,6)(5,12,14,11)(10,13,16,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,5)(2,10)(3,11)(4,12)(6,15)(7,14)(8,16)(9,13)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,37,39,34)(35,42,40,36)(38,41,46,45)(43,44,47,48);(1,6,7,9)(2,3,8,4)(5,13,14,15)(10,12,16,11)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,c^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, 8004117);