#6382787

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2726 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289983);