#6465731

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3118 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)(6,8)(9,10,14,19,15,20,25,30)(11,21,23,17,24,33,12,16)(13,18,29,36,28,35,37,39)(22,34,27,32,38,40,26,31);(2,4)(9,11,15,24)(10,16,20,17)(12,25,23,14)(13,27,28,26)(18,34,35,40)(19,33,30,21)(22,37,38,29)(31,39,32,36);(5,6,7,8)(9,12,14,11,15,23,25,24)(10,17,19,33,20,16,30,21)(13,26,29,22,28,27,37,38)(18,32,36,40,35,31,39,34);(9,13,14,29,15,28,25,37)(10,18,19,36,20,35,30,39)(11,22,23,27,24,38,12,26)(16,31,21,34,17,32,33,40);(13,28)(18,35)(22,38)(26,27)(29,37)(31,32)(34,40)(36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6372927);