#283800

order	512
nilpotency class	3
isoclinism	isoclinic class of #283771
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	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 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:

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

Polycyclic presentation:

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

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, 190996);