#209708

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

930 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)(6,10)(8,15)(11,17)(12,16)(14,19)(18,20)(21,22)(23,31)(24,29)(25,28)(26,30)(27,33)(32,36)(34,35);(5,6,9,12)(7,11,13,18)(8,16,15,10)(14,20,19,17)(21,23,26,34)(22,27,30,36)(24,32,25,33)(28,35,29,31);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,19)(16,20)(22,28)(24,25)(27,35)(29,30)(31,36)(32,33);(1,3,4,2)(7,13)(11,18)(14,19)(17,20)(21,24,26,25)(22,29,30,28)(23,32,34,33)(27,31,36,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116904);