#6465708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

3262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=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, 6372904);