#300375

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	#62535	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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)(4,14)(5,15)(6,10)(7,13)(9,11)(12,16)(17,18)(19,20)(21,32)(22,26)(23,27)(24,29)(25,31)(28,30);(1,3,11,4,6,12,5,13)(2,7,15,16,10,14,9,8)(17,18,22,32,23,27,21,26)(19,28,25,31,30,20,29,24);(3,12)(4,13)(7,14)(8,16)(17,19,21,29,23,30,22,25)(18,24,26,20,27,31,32,28);(1,4,11,12,6,13,5,3)(2,8,15,7,10,16,9,14)(17,20,21,31,23,28,22,24)(18,25,26,19,27,29,32,30)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,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,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, 207571);