#8445882

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=h,e^2=gi,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, 8353078);