#284410

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

710 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)(7,13)(8,16)(9,11)(10,14)(12,15)(17,20)(18,19)(21,22,25,23,26,29,27,30)(24,32,36,33,35,31,34,28);(1,2)(3,4)(6,11)(7,9)(12,16)(13,14)(15,18)(19,20)(21,23,25,29,26,30,27,22)(24,33,36,31,35,28,34,32);(2,4)(5,7,10,9)(6,11,14,13)(8,15,17,18)(12,16,19,20)(21,24,26,35)(22,28,29,33)(23,31,30,32)(25,34,27,36);(5,8,10,17)(6,12,14,19)(7,15,9,18)(11,16,13,20)(21,25,26,27)(22,23,29,30)(24,34,35,36)(28,31,33,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=hi,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,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, 191606);