#6466086

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442590
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320511, #72320615, #72320791
maximal normal subgroup of	#72320511, #72320615, #72320791
maximal quotient of	#72322914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373282);