#397916

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60423	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1933 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237646, #72251058, #72251087
maximal normal subgroup of	#72237646, #72251058, #72251087
maximal quotient of	#72314501

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305112);