#7726855

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7634051);