#474189

order	512
nilpotency class	3
isoclinism	isoclinic class of #457680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2665
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16904
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 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)(5,24,27,11)(6,28)(7,13)(8,12)(10,23)(15,44,46,36)(16,33)(17,34,29,35)(20,32,47,42)(21,54,25,52)(22,38)(26,40,31,41)(30,39)(37,56,57,55)(43,51)(45,62)(48,64)(49,61)(50,60)(53,63)(58,59);(1,3,7,17,8,18,6,29)(2,9,13,35,12,14,28,34)(4,20,30,15,22,47,23,46)(5,16,26,50,27,51,31,49)(10,36,19,32,39,44,38,42)(11,33,41,60,24,43,40,61)(21,53,55,45,25,58,56,48)(37,62,52,59,57,64,54,63);(1,4,6,30,8,22,7,23)(2,10,28,19,12,39,13,38)(3,15,17,20,18,46,29,47)(5,25,26,55,27,21,31,56)(9,32,35,36,14,42,34,44)(11,37,41,54,24,57,40,52)(16,48,49,53,51,45,50,58)(33,59,61,64,43,63,60,62);(1,5)(2,11)(3,16)(4,21)(6,31)(7,26)(8,27)(9,33)(10,37)(12,24)(13,41)(14,43)(15,45)(17,50)(18,51)(19,52)(20,53)(22,25)(23,56)(28,40)(29,49)(30,55)(32,59)(34,61)(35,60)(36,62)(38,54)(39,57)(42,63)(44,64)(46,48)(47,58)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381385);