#127314

order	512
nilpotency class	3
isoclinism	isoclinic class of #127312
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₄×M₁₆	C₂³
socle	C₂	C₂×#1288
prime radical	C₂⁴	#104
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

384 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)(3,11,12,23)(4,15,16,29)(5,9)(6,10)(7,13)(8,17,22,30)(14,25,27,31)(18,21)(19,26)(20,28)(24,32);(1,3)(2,7,9,20,10,19,21,24)(4,13)(5,14)(6,12)(8,11,15,25,22,23,29,31)(16,26)(17,28)(18,27)(30,32);(3,12)(7,19)(11,23)(13,26)(14,27)(20,24)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ghi,f^2=hi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,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	#72286532, #72286551, #72286560
maximal normal subgroup of	#72286532, #72286551, #72286560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34510);