#391930

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 914 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)(10,11)(14,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32);(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,28,22,20,29,27)(18,23,26,31,25,24,32,30);(3,7)(6,8)(9,11,13,10)(12,14,16,15)(17,20)(18,24)(19,22)(21,27)(23,25)(26,30)(28,29)(31,32);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(18,25)(23,24)(26,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299126);