#138066

order	512
nilpotency class	4
isoclinism	isoclinic class of #125393
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#392	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

848 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^d=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, 45262);