#8057928

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	#46080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965124);