#284545

order	512
nilpotency class	3
isoclinism	isoclinic class of #264283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^c=fi,g^c=gi,e^d=eh

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, 191741);