#243867

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
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	C₂×#399
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238928, #72238930, #72238947
maximal normal subgroup of	#72238928, #72238930, #72238947

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151063);