#440847

order	512
nilpotency class	3
isoclinism	isoclinic class of #440846
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 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)(3,14,18,9)(4,19,23,38)(5,24,26,11)(6,13)(7,29)(8,12)(10,22,37,30)(15,43)(16,39)(17,34,27,33)(20,42)(21,50)(25,41,28,40)(31,45)(32,49)(35,46)(36,52)(44,59,63,60)(47,58)(48,57)(51,56,64,62)(53,55)(54,61);(1,3,7,17,8,18,6,27)(2,9,29,33,12,14,13,34)(4,20,22,46,23,45,30,15)(5,16,25,48,26,49,28,47)(10,35,19,31,37,43,38,42)(11,39,40,57,24,32,41,58)(21,51,53,63,54,64,52,44)(36,60,50,62,55,59,61,56);(1,4,8,23)(2,10)(3,15,18,46)(5,21,26,54)(6,22,7,30)(9,31)(11,36)(12,37)(13,19)(14,42)(16,44,49,63)(17,45,27,20)(24,55)(25,52,28,53)(29,38)(32,56)(33,35)(34,43)(39,62)(40,61)(41,50)(47,51,48,64)(57,60)(58,59);(1,5)(2,11)(3,16)(4,21)(6,28)(7,25)(8,26)(9,32)(10,36)(12,24)(13,41)(14,39)(15,44)(17,48)(18,49)(19,50)(20,51)(22,53)(23,54)(27,47)(29,40)(30,52)(31,56)(33,58)(34,57)(35,59)(37,55)(38,61)(42,62)(43,60)(45,64)(46,63)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=ei,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,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319923, #72319988, #72320069
maximal normal subgroup of	#72319923, #72319988, #72320069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348043);