#317950

order	512
nilpotency class	3
isoclinism	isoclinic class of #317945
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

531 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

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

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

Polycyclic presentation:

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

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, 225146);