#494137

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

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,c^b=ci,d^c=di,g^c=gi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237213, #72239377, #72239832, …
maximal normal subgroup of	#72237213, #72239377, #72239832, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401333);