#251813

order	512
nilpotency class	3
isoclinism	isoclinic class of #251809
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

791 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 791 conjugacy classes of subgroups — too many to draw. Only its 134 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:

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

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,d^b=dhi,f^b=fh,e^c=eh,f^c=fi,g^c=gh,f^d=fh

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, 159009);