#7623243

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216687, #72216701, #72216707, …
maximal normal subgroup of	#72216687, #72216701, #72216707, …
maximal quotient of	C₂×#7623243, #72319432, #72321767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530439);