#401232

order	512
nilpotency class	3
isoclinism	isoclinic class of #394483
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

950 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)(3,14,18,33)(4,10)(5,24)(6,12)(7,28)(8,13)(9,17,32,27)(11,25)(15,41,19,39)(16,44,46,31)(20,50)(21,35)(22,36)(23,53)(26,37)(29,38)(30,49,56,43)(34,51)(40,45,57,47)(42,55,48,64)(52,58)(54,59)(60,63,61,62);(1,3,6,27,8,18,7,17)(2,9,12,33,13,32,28,14)(4,19,21,43,22,15,23,49)(5,16,25,47,26,46,29,45)(10,30,35,41,36,56,53,39)(11,31,37,57,38,44,24,40)(20,48,51,62,52,42,54,63)(34,55,58,61,59,64,50,60);(1,4,6,21,8,22,7,23)(2,10,12,35,13,36,28,53)(3,15,17,43,18,19,27,49)(5,20,25,51,26,52,29,54)(9,30,14,39,32,56,33,41)(11,34,37,58,38,59,24,50)(16,42,45,62,46,48,47,63)(31,55,40,60,44,64,57,61);(1,5,6,25,8,26,7,29)(2,11,12,37,13,38,28,24)(3,16,27,47,18,46,17,45)(4,20,21,51,22,52,23,54)(9,31,33,57,32,44,14,40)(10,34,35,58,36,59,53,50)(15,42,49,63,19,48,43,62)(30,55,41,61,56,64,39,60)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=ef,e^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^c=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, 308428);