#6382782

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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

2740 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)(11,17,23,16,25,31,24,32)(12,18,28,35,29,36,27,34)(22,33,37,30,38,40,26,39);(5,7,8,6)(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,20)(18,33)(19,31)(21,32)(22,27)(28,38)(29,37)(30,34)(35,39)(36,40);(1,2,3,4)(9,11)(10,17)(12,22)(13,24)(14,23)(15,25)(16,20)(18,33)(19,32)(21,31)(26,27)(28,37)(29,38)(30,35)(34,39)(36,40);(9,12,14,28,15,29,13,27)(10,18,20,35,21,36,19,34)(11,22,23,37,25,38,24,26)(16,30,31,40,32,39,17,33);(9,13,15,14)(10,19,21,20)(11,23,25,24)(12,27,29,28)(16,31,32,17)(18,34,36,35)(22,37,38,26)(30,40,39,33)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289978);