#351590

order	512
nilpotency class	3
isoclinism	isoclinic class of #347818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29806	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

708 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,5)(4,6)(7,8)(10,14)(11,16)(12,15)(18,22)(19,25)(20,28)(23,29)(24,31)(26,30);(1,3,4,7)(2,5,6,8)(9,10,15,16,13,11,12,14)(17,18,20,22,21,23,28,29)(19,26,27,31,25,24,32,30);(2,6)(5,8)(18,23)(22,29)(24,26)(30,31);(9,11,12,16,13,10,15,14)(17,19,20,27,21,25,28,32)(18,24,22,30,23,26,29,31)

Polycyclic presentation:

2^9|b^2=efi,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=chi,d^b=dh,g^d=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, 258786);