#8464392

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

4085 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

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, 8371588);