#206270

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
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×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 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)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20)(21,22)(23,31)(24,27)(25,35)(26,30)(28,33)(29,32)(34,36);(2,4)(5,7,8,15,10,17,9,16)(6,11,18,20,13,19,12,14)(21,23,32,34,26,33,25,24)(22,27,35,28,30,36,29,31);(5,6)(7,11)(8,18)(9,12)(10,13)(14,16)(15,20)(17,19)(21,24,25,33,26,34,32,23)(22,28,29,27,30,31,35,36);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)(23,33)(24,34)(27,36)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224715, #72224716, #72224717
maximal normal subgroup of	#72224715, #72224716, #72224717

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113466);