#284358

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,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, 191554);