#277033

order	512
nilpotency class	3
isoclinism	isoclinic class of #267051
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

722 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)(5,6,9,13,10,14,8,12)(7,15,16,19,17,20,18,11)(21,22,24,29,26,30,25,28)(23,27,32,35,33,34,31,36);(1,2,4,3)(21,23,26,33)(22,27,30,34)(24,32,25,31)(28,36,29,35);(5,7,10,17)(6,11,14,19)(8,16,9,18)(12,15,13,20)(22,28)(23,31)(24,25)(27,34)(29,30)(32,33);(1,3,4,2)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,15,19,20)

Polycyclic presentation:

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

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, 184229);