#432232

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15453
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1327 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)(3,8,15,29)(4,9)(5,21)(6,24)(7,11)(10,23)(12,26)(13,37,17,27)(14,40,41,54)(16,44,43,30)(18,48)(19,32)(20,52)(22,36)(25,35)(28,45,34,42)(31,50)(33,51)(38,61,46,64)(39,56,47,60)(49,59)(53,58)(55,62,57,63);(1,3)(2,8,11,29)(4,17)(5,14,22,41)(6,16)(7,15)(9,27,32,37)(10,34)(12,30,24,44)(13,19)(18,46,49,38)(20,47)(21,54)(23,42,25,45)(26,43)(28,35)(31,57)(33,56,52,60)(36,40)(39,51)(48,61)(50,63,53,62)(55,58)(59,64);(1,4)(2,9)(3,13,15,17)(5,18)(6,20)(7,19)(8,27,29,37)(10,31)(11,32)(12,33)(14,38,41,46)(16,39,43,47)(21,48)(22,49)(23,50)(24,52)(25,53)(26,51)(28,55,34,57)(30,56,44,60)(35,58)(36,59)(40,64,54,61)(42,62,45,63);(1,5)(2,10)(3,14,15,41)(4,18)(6,25)(7,22)(8,28,29,34)(9,31)(11,35)(12,36)(13,38,17,46)(16,45,43,42)(19,49)(20,53)(21,24)(23,26)(27,55,37,57)(30,54,44,40)(32,58)(33,59)(39,63,47,62)(48,52)(50,51)(56,61,60,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339428);