#6345499

order	512
nilpotency class	3
isoclinism	isoclinic class of #61591
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	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1580 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,d^b=di,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, 6252695);