#8446010

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	#42059	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,40,38)(35,45,47,41)(36,43,37,42)(39,44,48,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(18,19)(23,25)(24,26)(30,31)(33,35,40,47)(34,41,38,45)(36,46,37,44)(39,43,48,42);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,36,40,37)(34,42,38,43)(35,46,47,44)(39,45,48,41);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,37,40,36)(34,43,38,42)(35,46,47,44)(39,45,48,41);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,38,40,34)(35,45,47,41)(36,43,37,42)(39,46,48,44);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,39,40,48)(34,44,38,46)(35,43,47,42)(36,45,37,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,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, 8353206);