#7624084

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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1302 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^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.

Relationships

maximal subgroup of	#72321492, #72321618, #72321763, …
maximal normal subgroup of	#72321492, #72321618, #72321763, …
maximal quotient of	#72343697, #72343730

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531280);