#417659

order	512
nilpotency class	3
isoclinism	isoclinic class of #417622
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2425 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)(7,18)(8,13)(9,16)(10,15)(11,17)(12,21)(14,25)(19,32)(20,29)(22,30)(23,35)(24,33)(26,34)(27,31)(28,36);(5,7)(6,12)(8,23)(9,27)(10,21)(11,22)(13,32)(14,28)(15,30)(16,18)(17,31)(19,25)(20,24)(26,36)(29,34)(33,35);(2,4)(5,8)(6,13)(7,19)(9,24)(10,25)(11,26)(12,28)(14,15)(16,33)(17,34)(18,32)(20,22)(21,36)(23,27)(29,30)(31,35);(5,8)(6,14)(7,20)(9,25)(10,24)(11,26)(12,29)(13,15)(16,34)(17,33)(18,35)(19,22)(21,36)(23,27)(28,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324855);