#426280

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15349
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333476);