#8167395

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(33,34)(35,44)(36,37)(38,47)(39,43)(40,46)(41,45)(42,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,18,22,24)(19,20,27,25)(21,23,30,29)(26,28,32,31)(33,35,39,46)(34,40,43,44)(36,42,41,47)(37,38,45,48);(2,3)(7,9)(8,10)(14,15)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,36,39,41)(34,37,43,45)(35,42,46,47)(38,44,48,40);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20,22,25)(18,19,24,27)(21,28,30,31)(23,26,29,32)(33,37,39,45)(34,41,43,36)(35,38,46,48)(40,47,44,42);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,34)(35,40)(36,37)(38,42)(39,43)(41,45)(44,46)(47,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,23)(19,26)(20,28)(22,30)(24,29)(25,31)(27,32)(33,38)(34,42)(35,45)(36,44)(37,46)(39,48)(40,41)(43,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=gi,e^2=i,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, 8074591);