#8158700

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	#34589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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)(3,7)(4,12)(5,10)(6,9)(8,14)(11,16)(13,15)(18,22)(23,28)(24,29)(30,32)(33,34,38,39)(35,41,36,40)(37,45,46,42)(43,47,44,48);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(19,25)(23,28)(26,31)(30,32)(34,39)(40,41)(42,45)(47,48);(1,2,6,9)(3,10,5,7)(4,8,14,12)(11,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,2)(3,10)(4,12)(5,7)(6,9)(8,14)(11,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,34)(35,41)(36,40)(37,42)(38,39)(43,48)(44,47)(45,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,e^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, 8065896);