#244165

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

756 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151361);