#220898

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

853 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128094);