#215142

order	512
nilpotency class	3
isoclinism	isoclinic class of #213537
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1306 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1306 conjugacy classes of subgroups — too many to draw. Only its 234 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)(3,7)(4,6)(5,8)(9,10)(11,15,16,14)(12,13)(17,18,21,24,22,25,20,28)(19,23,27,32,26,30,29,31);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(19,26)(23,30)(27,29)(31,32);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,29)(21,27)(22,26)(24,32)(25,30)(28,31);(2,6)(5,7)(10,13)(14,15)(17,20,22,21)(18,24,25,28)(19,27,26,29)(23,31,30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^c=di,g^c=gh

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, 122338);