#498797

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319934, #72320798, #72320810, …
maximal normal subgroup of	#72319934, #72320798, #72320810, …
maximal quotient of	#72322649

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405993);