#384339

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1003 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291535);