#244141

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
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

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

normal subgroups only

This group has 770 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)(9,10,13,17,14,18,27,34)(11,19,21,28,22,33,26,15)(12,16,25,32,24,31,39,40)(20,35,36,30,23,37,38,29);(2,6)(3,4)(7,8)(9,11)(10,15)(12,23)(13,26)(14,22)(16,30)(17,33)(18,28)(19,34)(20,24)(21,27)(25,36)(29,31)(32,35)(37,40)(38,39);(2,6)(3,4)(7,8)(12,24)(15,28)(16,31)(19,33)(20,23)(25,39)(32,40)(36,38);(1,3,5,4)(2,6,8,7)(9,12,13,25,14,24,27,39)(10,16,17,32,18,31,34,40)(11,20,26,38,22,23,21,36)(15,29,33,37,28,30,19,35)

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=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, 151337);