#244149

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	#10032	C₂
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 150 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,d^2=efh,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, 151345);