#8505344

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

3844 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412540);