#244160

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
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

678 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=eh,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151356);