#6357801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60639	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2152 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)(9,10,15,19)(11,20,22,35)(12,17,26,33)(13,18,14,28)(16,23,30,24)(21,34,36,40)(25,31,27,32)(29,38,39,37);(9,11,14,24,15,22,13,23)(10,16,18,20,19,30,28,35)(12,21,25,37,26,36,27,38)(17,29,32,34,33,39,31,40);(1,2,3,5,4,6,7,8)(9,12)(10,17)(11,21)(13,27)(14,25)(15,26)(16,29)(18,32)(19,33)(20,34)(22,36)(23,38)(24,37)(28,31)(30,39)(35,40);(2,6)(5,8)(11,22)(16,30)(20,35)(21,36)(23,24)(29,39)(34,40)(37,38);(9,13,15,14)(10,18,19,28)(11,23,22,24)(12,25,26,27)(16,20,30,35)(17,31,33,32)(21,37,36,38)(29,40,39,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306813, #72306815, #72306816, …
maximal normal subgroup of	#72306813, #72306815, #72306816, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264997);