#7623812

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×#509
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#61234	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1288 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531008);