#495700

order	512
nilpotency class	4
isoclinism	isoclinic class of #2190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₄×#447
prime radical	#33530	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1029 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,f^b=fi,h^b=hi,e^c=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, 402896);