#362128

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13907
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269324);