#127388

order	512
nilpotency class	4
isoclinism	isoclinic class of #127185
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	C₂×C₄×D₄	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

616 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316687, #72316694, #72316707, …
maximal normal subgroup of	#72316687, #72316694, #72316707, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34584);