#6574019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481215);