#120027

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#386
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

654 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=h,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27223);