#420381

order	512
nilpotency class	3
isoclinism	isoclinic class of #420381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15277
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1295 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)(3,15,19,34)(4,10)(5,24,27,42)(6,12)(7,13)(8,14)(9,18,33,28)(11,26,41,30)(16,35,45,43)(17,32)(20,31,46,57)(21,50,53,55)(22,37)(23,39)(25,40)(29,38)(36,54,61,52)(44,56)(47,58)(48,59)(49,62)(51,64)(60,63);(1,3)(2,9)(4,20)(5,25)(6,28)(7,18)(8,19)(10,35)(11,40)(12,34)(13,15)(14,33)(16,22)(17,26)(21,51)(23,46)(24,32)(27,47)(29,45)(30,48)(31,38)(36,56)(37,57)(39,43)(41,58)(42,59)(44,53)(49,52)(50,62)(54,63)(55,60)(61,64);(1,4,6,22,8,23,7,29)(2,10,12,37,14,39,13,38)(3,16,18,20,19,45,28,46)(5,21,26,52,27,53,30,54)(9,31,15,43,33,57,34,35)(11,36,24,55,41,61,42,50)(17,44,25,49,48,51,47,63)(32,56,40,62,59,64,58,60);(1,5,6,26,8,27,7,30)(2,11,12,24,14,41,13,42)(3,17,18,25,19,48,28,47)(4,21,22,52,23,53,29,54)(9,32,15,40,33,59,34,58)(10,36,37,55,39,61,38,50)(16,44,20,49,45,51,46,63)(31,56,43,62,57,64,35,60)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264921, #72264928, #72264996
maximal normal subgroup of	#72264921, #72264928, #72264996
maximal quotient of	#72265601

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327577);