#209715

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	C₂×#555	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

982 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)(5,6,11,15)(7,16,8,13)(9,14,10,19)(12,18,20,17)(21,22)(23,31)(24,29)(25,28)(26,30)(27,33)(32,36)(34,35);(5,7,11,8)(6,12,15,20)(9,17,10,18)(13,19,16,14)(21,23,26,34)(22,27,30,36)(24,32,25,33)(28,35,29,31);(5,8,11,7)(6,13,15,16)(9,17,10,18)(12,19,20,14)(22,28)(24,25)(27,35)(29,30)(31,36)(32,33);(1,3,4,2)(5,9,11,10)(6,14,15,19)(7,17,8,18)(12,13,20,16)(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,c^2=hi,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, 116911);