#127589

order	512
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

978 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 154 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)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(6,9,11,10)(13,15)(14,19,21,27)(16,24,17,23)(18,25)(20,28)(22,26);(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34785);