#371872

order	512
nilpotency class	3
isoclinism	isoclinic class of #219498
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1107 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)(5,6,11,16,12,17,10,21)(7,8,20,23,19,24,14,13)(9,25,27,31,28,35,26,15)(18,32,34,36,33,29,30,22);(2,4)(5,7)(6,13,17,23)(8,21,24,16)(9,18)(10,14)(11,20)(12,19)(15,29,31,32)(22,35,36,25)(26,30)(27,34)(28,33);(2,4)(5,8,10,23,12,24,11,13)(6,14,21,7,17,20,16,19)(9,22,26,32,28,36,27,29)(15,30,35,18,31,34,25,33);(2,4)(5,9,11,27,12,28,10,26)(6,15,21,35,17,31,16,25)(7,18,20,34,19,33,14,30)(8,22,13,29,24,36,23,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226669, #72226674, #72226675
maximal normal subgroup of	#72226669, #72226674, #72226675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279068);