#238036

order	512
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1020 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 64

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

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

Polycyclic presentation:

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

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, 145232);