#277270

order	512
nilpotency class	3
isoclinism	isoclinic class of #267291
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

700 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)(5,6,8,12,9,11,17,19)(7,13,14,18,15,20,16,10)(21,22,24,28,26,30,25,29)(23,27,31,36,33,34,32,35);(1,2,4,3)(21,23,26,33)(22,27,30,34)(24,31,25,32)(28,36,29,35);(5,7,9,15)(6,10,11,18)(8,16,17,14)(12,20,19,13)(22,28)(23,31)(24,25)(27,34)(29,30)(32,33);(1,3,4,2)(6,11)(10,18)(12,19)(13,20)(21,24,26,25)(22,29,30,28)(23,31,33,32)(27,35,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 184466);