#6354765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354705
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	C₄²	C₂⁵
prime radical	C₂×#3035	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1782 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306793, #72306852
maximal normal subgroup of	#72306793, #72306852

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261961);