#251795

order	512
nilpotency class	3
isoclinism	isoclinic class of #251791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2404
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

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

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, 158991);