#382696

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2010 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 2010 conjugacy classes of subgroups — too many to draw. Only its 282 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,25)(21,27)(23,30)(24,29)(26,28)(31,32);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(19,25,27,26)(22,29,30,31);(3,7)(6,8)(18,23)(22,30)(24,32)(29,31)

Polycyclic presentation:

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

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, 289892);