#423436

order	512
nilpotency class	3
isoclinism	isoclinic class of #423435
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15314
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

1268 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)(3,9)(4,10,21,35)(5,24)(6,13)(7,28)(8,12)(11,26)(14,39,41,29)(15,43)(16,32)(17,31)(18,47)(19,48,42,33)(20,50,52,62)(22,36,23,55)(25,37)(27,38)(30,46)(34,53,61,54)(40,59,64,60)(44,58)(45,57)(49,63,51,56);(1,3)(2,9)(4,19,21,42)(5,15)(6,17)(7,18)(8,16)(10,33,35,48)(11,30)(12,32)(13,31)(14,22,41,23)(20,51,52,49)(24,43)(25,45)(26,46)(27,44)(28,47)(29,36,39,55)(34,60,61,59)(37,57)(38,58)(40,53,64,54)(50,63,62,56);(1,4)(2,10,12,35)(3,14,16,41)(5,20)(6,22)(7,23)(8,21)(9,29)(11,34,37,61)(13,36,28,55)(15,40,44,64)(17,19,18,42)(24,50,38,62)(25,53)(26,54)(27,52)(30,56)(31,33)(32,39)(43,60)(45,51,46,49)(47,48)(57,63)(58,59);(1,5,8,27)(2,11,12,37)(3,15,16,44)(4,20,21,52)(6,26,7,25)(9,30,32,57)(10,34,35,61)(13,38,28,24)(14,40,41,64)(17,46,18,45)(19,49,42,51)(22,54,23,53)(29,56,39,63)(31,58,47,43)(33,59,48,60)(36,62,55,50)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306392, #72306450, #72306455
maximal normal subgroup of	#72306392, #72306450, #72306455

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330632);