#391878

order	512
nilpotency class	3
isoclinism	isoclinic class of #213555
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1180 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,30)(21,24)(22,25)(26,28)(27,32)(29,31);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,29,22,28,20,27)(18,23,24,31,25,26,30,32);(3,7)(6,8)(10,14)(13,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,h^c=hi

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, 299074);