#136005

order	512
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

561 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=ei,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, 43201);