#8634173

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	#45522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8541369);