#417488

order	512
nilpotency class	3
isoclinism	isoclinic class of #417487
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

2421 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)(7,17)(8,20)(9,13)(10,15)(11,27)(12,16)(14,24)(18,31)(19,29)(21,34)(22,36)(23,33)(25,32)(26,30)(28,35);(1,2)(3,4)(5,7)(6,13)(8,21)(9,10)(11,19)(12,26)(14,31)(15,17)(16,29)(18,25)(20,28)(22,24)(23,35)(27,30)(32,36)(33,34);(5,8)(6,14)(7,18)(9,22)(10,23)(11,25)(12,24)(13,28)(15,32)(16,33)(17,34)(19,35)(20,27)(21,26)(29,31)(30,36);(2,4)(5,9,10,7)(6,13,15,17)(8,22,23,18)(11,19,12,26)(14,28,32,34)(16,30,27,29)(20,31,33,36)(21,25,35,24)

Polycyclic presentation:

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

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, 324684);