#6490948

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470028
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

3111 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:

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

Polycyclic presentation:

2^9|c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,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, 6398144);