#137318

order	512
nilpotency class	4
isoclinism	isoclinic class of #12148
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×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

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

normal subgroups only

This group has 697 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=gi,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=di,e^a=egh,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi

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, 44514);