#127223

order	512
nilpotency class	4
isoclinism	isoclinic class of #127215
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gh,e^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,d^c=di,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237048, #72238399, #72238400, …
maximal normal subgroup of	#72237048, #72238399, #72238400, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34419);