#381887

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
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	#23426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1243 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 289083);