#287817

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6,10,14,9,13,16,19)(7,15,17,12,8,11,18,20)(21,22,24,28,25,29,30,27)(23,26,32,33,31,34,36,35);(1,2,4,3)(5,7,16,18,9,8,10,17)(6,11,19,12,13,15,14,20)(21,23,30,36,25,31,24,32)(22,26,27,35,29,34,28,33);(5,8,9,7)(6,12,13,20)(10,17,16,18)(11,14,15,19)(22,27)(23,31)(24,30)(26,33)(28,29)(34,35);(1,3,4,2)(6,13)(11,15)(12,20)(14,19)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=hi,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195013);