#8359928

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	#38325	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8267124);