#7955138

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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)(9,10,15,19)(11,18,13,16)(12,21,14,17)(20,23,22,24)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,19,18)(12,22,14,20)(17,24,21,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,4,5,2)(3,7,8,6)(9,12,15,14)(10,17,19,21)(11,20,13,22)(16,23,18,24)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,13,15,11)(10,18,19,16)(12,20,14,22)(17,23,21,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,39)(38,40);(9,10)(11,16)(12,17)(13,18)(14,21)(15,19)(20,23)(22,24)(25,29,30,38)(26,32,33,37)(27,34,35,40)(28,36,31,39);(9,14,15,12)(10,17,19,21)(11,20,13,22)(16,24,18,23)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=i,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, 7862334);