#317947

order	512
nilpotency class	3
isoclinism	isoclinic class of #317945
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 557 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:

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

Polycyclic presentation:

2^9|b^2=eg,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, 225143);