#99184

order	512
nilpotency class	3
isoclinism	isoclinic class of #99180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

474 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,f^b=fg,e^c=eh,f^c=fg

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, 6380);