#8445946

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	#37214	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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)(17,18,24,27)(19,25,22,23)(20,30,21,26)(28,31,29,32)(33,34,40,38)(35,45,47,41)(36,43,37,42)(39,44,46,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,19,24,22)(18,25,27,23)(20,29,21,28)(26,31,30,32)(33,35)(34,41)(36,46)(37,39)(38,45)(40,47)(42,48)(43,44);(17,20,24,21)(18,26,27,30)(19,28,22,29)(23,31,25,32)(33,36,40,37)(34,42,38,43)(35,46,47,39)(41,48,45,44);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,21,24,20)(18,26,27,30)(19,29,22,28)(23,31,25,32)(33,37,40,36)(34,43,38,42)(35,46,47,39)(41,48,45,44);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,22,24,19)(18,23,27,25)(20,29,21,28)(26,31,30,32)(33,38,40,34)(35,45,47,41)(36,43,37,42)(39,48,46,44);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,23)(18,19)(20,31)(21,32)(22,27)(24,25)(26,28)(29,30)(33,39,40,46)(34,44,38,48)(35,37,47,36)(41,43,45,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,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, 8353142);