#282711

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

737 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189907);