#6344455

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=ei,d^c=di,e^c=eh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251651);