#8445936

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(17,18,23,26)(19,22,25,20)(21,29,30,24)(27,28,32,31)(33,34,38,42)(35,37,41,39)(36,44,47,40)(43,46,45,48);(1,3,8,6)(2,9,11,7)(4,13,5,12)(10,15,14,16)(17,19,23,25)(18,20,26,22)(21,27,30,32)(24,28,29,31)(33,35,38,41)(34,39,42,37)(36,45,47,43)(40,48,44,46);(1,4,8,5)(2,10,11,14)(3,12,6,13)(7,15,9,16)(17,20,23,22)(18,19,26,25)(21,28,30,31)(24,27,29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,8,4)(2,10,11,14)(3,13,6,12)(7,15,9,16)(17,21)(18,24)(19,27)(20,28)(22,31)(23,30)(25,32)(26,29)(33,36,38,47)(34,40,42,44)(35,43,41,45)(37,48,39,46);(1,6,8,3)(2,7,11,9)(4,13,5,12)(10,15,14,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,32);(1,7)(2,3)(4,15)(5,16)(6,11)(8,9)(10,12)(13,14)(17,22,23,20)(18,25,26,19)(21,28,30,31)(24,27,29,32)(33,37,38,39)(34,41,42,35)(36,46,47,48)(40,43,44,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=h,e^2=i,f^2=g,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, 8353132);