#7715537

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	C₂⁵	C₂⁴
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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622733);