#277102

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 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)(21,22,26,28,25,29,24,30)(23,31,32,36,34,27,33,35);(1,2,4,3)(21,23,25,34)(22,27,29,31)(24,33,26,32)(28,35,30,36);(5,7,10,17)(6,11,14,19)(8,16,9,18)(12,15,13,20)(22,28)(23,32)(24,26)(27,31)(29,30)(33,34);(1,3,4,2)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,15,19,20)(21,24,25,26)(22,28,29,30)(23,33,34,32)(27,35,31,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,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,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184298);