#8445904

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	#45982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=h,e^2=i,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, 8353100);