#8406876

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	#57199	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314072);