#106741

order	512
nilpotency class	3
isoclinism	isoclinic class of #106738
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1030 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=eh,d^c=dh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277968, #72278022, #72278163
maximal normal subgroup of	#72277968, #72278022, #72278163

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13937);