#314967

order	512
nilpotency class	3
isoclinism	isoclinic class of #251800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2487
prime radical	#392	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

546 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)(3,5)(4,7)(6,8)(9,10,15,18,14,19,13,27)(11,20,28,31,21,34,23,16)(12,24,29,32,25,38,26,17)(22,36,39,40,35,30,37,33);(9,11,13,23,14,21,15,28)(10,16,18,20,19,31,27,34)(12,22,26,37,25,35,29,39)(17,30,24,33,32,36,38,40);(2,5)(3,8)(6,7)(11,21)(12,25)(17,32)(23,28)(24,38)(26,29)(30,36)(33,40);(9,12,13,26,14,25,15,29)(10,17,18,24,19,32,27,38)(11,22,23,37,21,35,28,39)(16,30,20,33,31,36,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222163);