#6346200

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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	#61450	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1566 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18,21,25,23,19,22,27)(20,26,29,31,30,24,28,32);(1,3,7,5)(2,8,11,16)(4,13,6,14)(9,10,15,12)(17,19,23,18)(20,26,30,24)(21,27,22,25)(28,32,29,31);(1,4,7,6)(2,9,11,15)(3,13,5,14)(8,10,16,12)(17,20,22,28,23,30,21,29)(18,24,27,31,19,26,25,32);(1,5,7,3)(2,10,11,12)(4,13,6,14)(8,15,16,9)(17,21,23,22)(18,25,19,27)(20,28,30,29)(24,31,26,32);(1,4,7,6)(2,9,11,15)(3,13,5,14)(8,10,16,12)(17,21,23,22)(18,25,19,27)(20,29,30,28)(24,32,26,31)

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=ci,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, 6253396);