#300369

order	512
nilpotency class	3
isoclinism	isoclinic class of #300333
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	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 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)(3,11)(4,7)(5,14)(6,8)(9,12)(10,15)(13,16)(17,18)(19,21)(20,22)(23,27)(24,28)(25,29)(26,32)(30,31);(1,3,5,4,6,12,15,13)(2,7,10,11,8,16,14,9)(17,19,22,29,24,30,23,26)(18,25,27,21,28,32,20,31);(2,8)(3,12)(4,13)(10,14)(17,20,22,18,24,27,23,28)(19,25,29,21,30,32,26,31);(1,4,5,12,6,13,15,3)(2,9,10,7,8,11,14,16)(17,21,23,25,24,31,22,32)(18,26,20,30,28,29,27,19)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=ei,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,d^b=dh,g^b=gi,d^c=dhi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207565);