#494132

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,d^b=di,g^c=gi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237213, #72239217, #72239374, …
maximal normal subgroup of	#72237213, #72239217, #72239374, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401328);