#244178

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
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

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

normal subgroups only

This group has 750 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:

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

Polycyclic presentation:

2^9|a^2=e,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=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151374);