#7927670

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	#42533	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834866);