#404536

order	512
nilpotency class	3
isoclinism	isoclinic class of #395191
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14893
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

977 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285525, #72285582, #72286143, …
maximal normal subgroup of	#72285525, #72285582, #72286143, …
maximal quotient of	C₂×#404536

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311732);