#6343108

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2867 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250304);