#94324

order	512
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₄²
Frattini, cosocle	C₂×#384	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#384 ◁ G
LECS	G ▷ C₂×#384 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

365 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,e^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,h^a=hi,c^b=cg,g^b=gh,e^c=ehi,g^c=gi

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, 1520);