#243908

order	512
nilpotency class	3
isoclinism	isoclinic class of #232766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

751 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283833, C₂×#243908, #72307163, …
maximal normal subgroup of	#72283833, C₂×#243908, #72307163, …
maximal quotient of	C₂×#243908

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151104);