#8720773

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627969);