#8533152

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440348);