#401235

order	512
nilpotency class	3
isoclinism	isoclinic class of #394481
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14814
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15,19,33)(4,10)(5,25)(6,12)(7,13)(8,14)(9,18,32,28)(11,26)(16,34,44,41)(17,45,47,31)(20,30,50,56)(21,51)(22,36)(23,37)(24,38)(27,39)(29,40)(35,52)(42,46,57,48)(43,55,62,64)(49,61,63,58)(53,59)(54,60);(1,3,6,28,8,19,7,18)(2,9,12,33,14,32,13,15)(4,20,22,16,24,50,23,44)(5,17,26,48,27,47,29,46)(10,34,36,56,38,41,37,30)(11,31,39,57,40,45,25,42)(21,49,52,43,53,63,54,62)(35,58,59,64,60,61,51,55);(1,4,6,22,8,24,7,23)(2,10,12,36,14,38,13,37)(3,16,18,20,19,44,28,50)(5,21,26,52,27,53,29,54)(9,30,15,41,32,56,33,34)(11,35,39,59,40,60,25,51)(17,43,46,49,47,62,48,63)(31,55,42,61,45,64,57,58);(1,5,6,26,8,27,7,29)(2,11,12,39,14,40,13,25)(3,17,28,48,19,47,18,46)(4,21,22,52,24,53,23,54)(9,31,33,57,32,45,15,42)(10,35,36,59,38,60,37,51)(16,43,50,63,44,62,20,49)(30,55,34,58,56,64,41,61)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=ef,e^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,e^b=ei,f^b=fi,g^b=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308431);