#8037552

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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)(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(28,36)(31,38)(34,40)(37,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(10,15)(11,18)(16,21)(19,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,39)(33,38)(35,40);(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,f^2=g,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, 7944748);