#8720427

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=i,e^2=h,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, 8627623);