#417531

order	512
nilpotency class	3
isoclinism	isoclinic class of #417530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2465 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)(7,12)(8,18)(9,14)(10,23)(11,15)(13,16)(17,26)(19,31)(20,25)(21,34)(22,29)(24,28)(27,30)(32,36)(33,35);(5,7,11,10)(6,12,15,23)(8,16,20,24)(9,17,22,21)(13,18,28,25)(14,26,29,34)(19,32,33,30)(27,35,36,31);(5,8)(6,13)(7,16)(9,19)(10,24)(11,20)(12,25)(14,27)(15,28)(17,30)(18,23)(21,32)(22,33)(26,35)(29,36)(31,34);(2,4)(5,9)(6,14)(7,17)(8,19)(10,21)(11,22)(12,26)(13,27)(15,29)(16,30)(18,31)(20,33)(23,34)(24,32)(25,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313070, #72313145, #72313151
maximal normal subgroup of	#72313070, #72313145, #72313151

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324727);