#245083

order	512
nilpotency class	3
isoclinism	isoclinic class of #245063
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1698 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1698 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152279);