#94275

order	512
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#353
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₂×#390	C₂²
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#390 ◁ G
LECS	G ▷ C₂×#390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 494 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) = 128

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

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

Polycyclic presentation:

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

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, 1471);