#8159076

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=g,e^2=gh,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^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, 8066272);