#6438028

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4598 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)(9,10,14,19,16,21,15,20)(11,22,24,32,12,26,25,17)(13,18,29,34,28,33,30,35)(23,36,37,40,27,39,38,31);(2,4)(5,6)(7,8)(9,11,16,12)(10,17,21,32)(13,23,28,27)(14,25,15,24)(18,31,33,40)(19,26,20,22)(29,38,30,37)(34,39,35,36);(6,8)(9,12,16,11)(10,17,21,32)(13,27,28,23)(14,24,15,25)(18,31,33,40)(19,26,20,22)(29,37,30,38)(34,39,35,36);(6,8)(9,13)(10,18)(11,23)(12,27)(14,29)(15,30)(16,28)(17,31)(19,34)(20,35)(21,33)(22,36)(24,37)(25,38)(26,39)(32,40);(13,28)(18,33)(23,27)(29,30)(31,40)(34,35)(36,39)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345224);