#8460666

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=gi,f^2=hi,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, 8367862);