#8464165

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	#53481	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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)(3,7)(4,14)(5,9)(6,10)(8,11)(12,16)(13,15)(17,18)(19,24)(20,25)(21,31)(22,27)(23,28)(26,30)(29,32)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3,8,6)(2,7,11,10)(4,13,5,12)(9,16,14,15)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,31)(34,39)(35,42)(41,44)(43,48);(1,4,8,5)(2,9,11,14)(3,12,6,13)(7,15,10,16)(17,19,23,32)(18,24,28,29)(20,30,27,21)(22,31,25,26)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,8,4)(2,9,11,14)(3,13,6,12)(7,15,10,16)(17,20)(18,25)(19,30)(21,32)(22,28)(23,27)(24,26)(29,31)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,6,8,3)(2,10,11,7)(4,13,5,12)(9,16,14,15)(17,21,23,30)(18,26,28,31)(19,20,32,27)(22,29,25,24)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,7,8,10)(2,3,11,6)(4,15,5,16)(9,12,14,13)(17,22,23,25)(18,27,28,20)(19,31,32,26)(21,29,30,24)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=h,e^2=i,f^2=i,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, 8371361);