#244166

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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

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

normal subgroups only

This group has 670 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)(9,10,15,27,14,19,13,18)(11,20,26,16,23,33,22,29)(12,17,28,38,24,31,25,32)(21,34,37,30,35,40,36,39);(2,6)(3,4)(7,8)(9,11)(10,16)(12,21)(13,26)(14,23)(15,22)(17,30)(18,33)(19,29)(20,27)(24,35)(25,37)(28,36)(31,39)(32,40)(34,38);(2,6)(3,4)(7,8)(12,24)(16,29)(17,31)(20,33)(21,35)(25,28)(32,38)(36,37);(1,3,5,4)(2,6,8,7)(9,12,13,25,14,24,15,28)(10,17,18,32,19,31,27,38)(11,21,26,37,23,35,22,36)(16,30,33,40,29,39,20,34)

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=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, 151362);