#6346164

order	512
nilpotency class	3
isoclinism	isoclinic class of #61559
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	#61434	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253360);