#6343194

order	512
nilpotency class	3
isoclinism	isoclinic class of #61591
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	#62816	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2905 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250390);