#197927

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3217	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105123);