#498792

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
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1600 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)(9,10,14,19)(11,16,20,30)(12,15,25,29)(13,26,24,18)(17,33,28,23)(21,27,34,38)(22,37,35,32)(31,40,39,36);(5,6)(7,8)(10,15)(12,23)(13,24)(16,27)(17,18)(19,29)(21,36)(22,35)(25,33)(26,28)(30,38)(31,32)(34,40)(37,39);(2,4)(6,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,27)(17,31)(18,32)(19,30)(23,36)(24,35)(25,34)(26,37)(28,39)(29,38)(33,40);(11,20)(16,30)(21,34)(22,35)(27,38)(31,39)(32,37)(36,40)

Polycyclic presentation:

2^9|a^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,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, #72320788, #72320837
maximal normal subgroup of	#72319934, #72320788, #72320837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405988);