#127242

order	512
nilpotency class	4
isoclinism	isoclinic class of #127217
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

805 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gh,e^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cgi,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	#72233278, #72233308, #72233312, …
maximal normal subgroup of	#72233278, #72233308, #72233312, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34438);