#129746

order	512
nilpotency class	4
isoclinism	isoclinic class of #124826
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1158 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,21)(16,24,25,19)(17,20)(18,22,26,28)(23,27);(6,9)(7,12)(10,11)(13,15,17,23)(14,18,20,26)(16,22,25,28)(19,21,24,27);(2,4)(13,16)(14,19)(15,22)(17,25)(18,21)(20,24)(23,28)(26,27)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,f^b=fgh,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293049, #72293137, #72293143
maximal normal subgroup of	#72293049, #72293137, #72293143

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36942);