#269754

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

910 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^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, 176950);