#285041

order	512
nilpotency class	3
isoclinism	isoclinic class of #284987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

642 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fi,g^c=ghi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192237);