#8445945

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	#54419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,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, 8353141);