#394046

order	512
nilpotency class	3
isoclinism	isoclinic class of #394046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4016 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282010
maximal normal subgroup of	#72282010
maximal quotient of	#72282260

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301242);