#417623

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

2369 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)(8,20)(9,10)(13,28)(14,23)(16,27)(17,32)(19,24)(21,22)(25,31)(29,30)(33,36)(34,35);(5,6)(7,16)(8,13)(9,23)(10,14)(11,15)(12,19)(17,33)(18,27)(20,28)(21,30)(22,29)(24,26)(25,32)(31,34)(35,36);(2,4)(5,7)(6,12)(8,17)(9,24)(10,19)(11,18)(13,25)(14,27)(15,26)(16,23)(20,32)(21,35)(22,34)(28,31)(29,36)(30,33);(5,8,11,22)(6,13,15,29)(7,17,18,34)(9,20,10,21)(12,25,26,36)(14,30,23,28)(16,31,27,33)(19,35,24,32)

Polycyclic presentation:

2^9|d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,g^d=gi,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, 324819);