#244142

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
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	#2397
prime radical	#2397	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 684 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151338);