#206209

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	C₂×#3207	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1156 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)(23,30)(24,28)(25,34)(26,29)(27,33)(31,36)(32,35);(2,4)(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16)(23,31)(27,35)(30,36)(32,33);(5,7)(6,12)(8,15)(9,14)(10,16)(11,18)(13,19)(17,20)(21,23,25,33,26,31,24,32)(22,27,28,30,29,35,34,36);(7,14)(12,19)(15,18)(16,20)(21,24,26,25)(22,28,29,34)(23,32,31,33)(27,30,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=i,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.

Relationships

maximal subgroup of	#72224715, #72224721
maximal normal subgroup of	#72224715, #72224721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113405);