#93492

order	512
nilpotency class	4
isoclinism	isoclinic class of #7316
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1304
commutator	C₂⁴	C₄×C₈
Frattini, cosocle	C₂³×M₁₆	C₂²
socle	C₂	#7032
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=h,h^2=i,b^a=bc,c^a=cf,e^a=eg,g^a=gi,c^b=cg,d^b=df,f^b=fi,e^d=ei

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, 688);