#456638

order	512
nilpotency class	3
isoclinism	isoclinic class of #456638
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

1361 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)(3,14)(4,10)(5,23,26,41)(6,28)(7,12)(8,13)(9,17)(11,27,40,25)(15,42)(16,47,50,32)(18,33)(19,52)(20,54,55,39)(21,38)(22,37)(24,58,56,35)(29,36)(30,34)(31,46)(43,49,57,51)(44,63,53,60)(45,61)(48,62,59,64);(1,3,6,17,8,18,7,30)(2,9,12,14,13,34,28,33)(4,19,21,46,22,15,29,45)(5,16,27,51,26,50,25,49)(10,31,36,52,37,61,38,42)(11,32,41,43,40,47,23,57)(20,53,56,48,55,44,24,59)(35,60,54,64,58,63,39,62);(1,4)(2,10)(3,15)(5,24)(6,29)(7,21)(8,22)(9,31)(11,39)(12,38)(13,37)(14,42)(16,48)(17,46)(18,19)(20,25)(23,35)(26,56)(27,55)(28,36)(30,45)(32,62)(33,52)(34,61)(40,54)(41,58)(43,63)(44,49)(47,64)(50,59)(51,53)(57,60);(1,5)(2,11)(3,16)(4,20)(6,27)(7,25)(8,26)(9,32)(10,35)(12,41)(13,40)(14,43)(15,44)(17,51)(18,50)(19,53)(21,56)(22,55)(23,28)(24,29)(30,49)(31,60)(33,57)(34,47)(36,54)(37,58)(38,39)(42,62)(45,59)(46,48)(52,64)(61,63)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287286, #72287302, #72287345
maximal normal subgroup of	#72287286, #72287302, #72287345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363834);