#135786

order	512
nilpotency class	4
isoclinism	isoclinic class of #12525
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=eh,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210384
maximal normal subgroup of	#72210384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42982);