#6343091

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62781	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2879 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343110
maximal normal subgroup of	#72343110
maximal quotient of	#72343174

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250287);