#8158896

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	#54135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,4)(5,13)(6,7)(8,12)(9,14)(10,16)(11,15)(17,18,23,26)(19,25,22,24)(20,21,30,32)(27,31,29,28)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,3,6,8)(2,4,7,12)(5,10,14,15)(9,11,13,16)(17,19,23,22)(18,24,26,25)(20,27,30,29)(21,28,32,31)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(2,7)(3,8)(9,13)(10,15)(17,20)(18,21)(19,27)(22,29)(23,30)(24,28)(25,31)(26,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,16)(14,15)(17,21,23,32)(18,20,26,30)(19,28,22,31)(24,27,25,29)(33,36,38,42)(34,39,41,35)(37,44,46,48)(40,47,45,43);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,13)(14,16)(17,20)(18,21)(19,27)(22,29)(23,30)(24,28)(25,31)(26,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,5,6,14)(2,9,7,13)(3,10,8,15)(4,11,12,16)(17,22,23,19)(18,25,26,24)(20,29,30,27)(21,31,32,28)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=hi,d^2=g,f^2=gh,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^d=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, 8066092);