#300101

order	512
nilpotency class	3
isoclinism	isoclinic class of #300077
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 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)(3,8)(4,13)(5,15)(6,10)(7,12)(9,14)(11,16)(17,18,24,28)(19,29,32,21)(20,22,27,23)(25,26,30,31);(1,3,5,12,6,11,14,4)(2,7,9,8,10,13,15,16)(17,19,22,31,24,32,23,26)(18,25,20,29,28,30,27,21);(3,11)(4,12)(7,13)(8,16)(17,20,23,18,24,27,22,28)(19,30,26,29,32,25,31,21);(1,4,14,11,6,12,5,3)(2,8,15,7,10,16,9,13)(17,21,23,30,24,29,22,25)(18,26,27,32,28,31,20,19)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=eh,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=dhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207297);