#149400

order	512
nilpotency class	4
isoclinism	isoclinic class of #13502
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11378
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

435 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=h,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=ghi,h^c=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, 56596);