#244159

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
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

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

normal subgroups only

This group has 764 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)(9,10,13,17,14,18,26,33)(11,19,21,29,22,35,28,15)(12,23,27,39,25,16,31,34)(20,37,38,30,24,36,40,32);(2,6)(3,4)(7,8)(9,11)(10,15)(12,24)(13,28)(14,22)(16,32)(17,35)(18,29)(19,33)(20,25)(21,26)(23,30)(27,38)(31,40)(34,36)(37,39);(2,7)(3,4)(6,8)(12,25)(15,29)(16,23)(19,35)(20,24)(27,31)(34,39)(38,40);(1,3,5,4)(2,6,8,7)(9,12,26,31,14,25,13,27)(10,16,33,39,18,23,17,34)(11,20,21,38,22,24,28,40)(15,30,19,36,29,32,35,37)

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=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, 151355);