#456663

order	512
nilpotency class	3
isoclinism	isoclinic class of #456662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16284
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286186, #72286206, #72286231
maximal normal subgroup of	#72286186, #72286206, #72286231

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363859);