#6346049

order	512
nilpotency class	3
isoclinism	isoclinic class of #61058
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×C₄×D₄
prime radical	#61445	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1573 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:

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

Polycyclic presentation:

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

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, 6253245);