#262990

order	512
nilpotency class	3
isoclinism	isoclinic class of #251800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2408
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 170186);