#115462

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22658);