#6435058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432876
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318405
maximal normal subgroup of	#72318405

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342254);