#471102

order	512
nilpotency class	3
isoclinism	isoclinic class of #459299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2752
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16751
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

519 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378298);