#8127632

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	#57210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=gh,e^2=g,f^2=h,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^c=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, 8034828);