#136236

order	512
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

607 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 88 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)(6,11)(7,12)(8,9)(13,14,17,21)(15,19,22,28)(16,18,24,20)(23,26,25,27);(1,3,4,2)(5,6,10,7)(8,12,9,11)(14,18)(15,22)(16,24)(19,27)(20,21)(26,28);(5,7,10,6)(8,12,9,11)(13,15,17,22)(14,19,21,28)(16,25,24,23)(18,26,20,27)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=g,d^2=g,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=egi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43432);