#118044

order	512
nilpotency class	3
isoclinism	isoclinic class of #10038
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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

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 — 3 generators on 36 points:

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

Polycyclic presentation:

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

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, 25240);