#206203

order	512
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1070 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)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,35)(34,36);(2,4)(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16)(22,26)(23,30)(28,33)(31,36);(5,7)(6,12)(8,15)(9,14)(10,16)(11,18)(13,19)(17,20)(21,23,25,31,24,30,32,36)(22,27,28,29,26,34,33,35);(7,14)(12,19)(15,18)(16,20)

Polycyclic presentation:

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

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, 113399);