#300073

order	512
nilpotency class	3
isoclinism	isoclinic class of #300013
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62532	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207269);