#6345255

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
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	#60944	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1584 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,16)(14,15)(17,18,22,25,23,19,21,26)(20,24,28,32,29,27,30,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,29)(28,31)(30,32);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,15)(11,16)(17,20,22,28,23,29,21,30)(18,24,25,32,19,27,26,31);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)(17,21,23,22)(18,25,19,26)(20,28,29,30)(24,31,27,32);(1,4,6,5)(2,9,11,10)(3,12,13,14)(7,8,15,16)(17,21,23,22)(18,26,19,25)(20,28,29,30)(24,32,27,31)

Polycyclic presentation:

2^9|a^2=g,c^2=g,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=dh,e^c=eh

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, 6252451);