#8167403

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=i,e^2=gi,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, 8074599);