#6435072

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432949
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3480 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)(11,15)(12,16)(13,25)(14,17)(18,26)(19,30)(20,34)(21,28)(22,27)(23,37)(24,31)(29,35)(32,38)(33,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,30)(17,28)(18,29)(19,24)(23,36)(25,34)(26,35)(27,31)(32,40)(33,38)(37,39);(2,5)(6,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,31)(18,32)(20,33)(21,22)(25,37)(26,38)(28,30)(29,39)(34,40)(35,36);(1,3,4,7)(2,6,5,8)(10,17)(15,28)(16,31)(18,25)(27,30)(29,34)(32,37)(39,40);(9,13,14,26)(10,18,17,25)(11,20,21,35)(12,23,24,38)(15,29,28,34)(16,32,31,37)(19,33,22,36)(27,39,30,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284184, #72284505, #72285237, …
maximal normal subgroup of	#72284184, #72284505, #72285237, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342268);