#282706

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	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

742 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)(5,6,10,14)(7,13,9,11)(8,16,17,20)(12,18,19,15)(23,30)(27,35)(31,36)(32,34);(1,3,4,2)(6,11)(7,9)(12,16)(13,14)(15,18)(19,20)(21,22,25,29)(23,31,30,36)(24,28,26,33)(27,34,35,32);(2,3)(5,7,10,9)(6,11,14,13)(8,15,17,18)(12,16,19,20)(21,23)(22,27)(24,32)(25,30)(26,34)(28,36)(29,35)(31,33);(5,8,10,17)(6,12,14,19)(7,15,9,18)(11,16,13,20)(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=fh,c^2=h,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, 189902);