#203112

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=di,d^c=dh,f^c=fhi

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, 110308);