#8167444

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8074640);