#456652

order	512
nilpotency class	3
isoclinism	isoclinic class of #456651
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	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287286, #72287312, #72287345
maximal normal subgroup of	#72287286, #72287312, #72287345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363848);