#499416

order	512
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2085
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2085
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

527 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)(3,15)(4,21,26,38)(5,11)(6,12)(7,29)(8,14)(9,17)(10,24,40,23)(13,32)(16,33,48,61)(18,50)(19,36)(20,34)(22,55,56,37)(25,58,59,44)(27,42)(28,41)(30,35)(31,39,46,64)(43,52)(45,47,60,53)(49,63)(51,54,57,62);(1,3,5,17,8,20,6,19)(2,9,12,15,14,36,11,34)(4,22,23,51,26,56,24,57)(7,30,27,18,32,49,28,52)(10,37,21,54,40,55,38,62)(13,43,42,63,29,50,41,35)(16,46,47,59,48,31,53,25)(33,44,45,64,61,58,60,39);(1,4,5,23,8,26,6,24)(2,10,12,21,14,40,11,38)(3,16,17,47,20,48,19,53)(7,31,27,25,32,46,28,59)(9,33,15,45,36,61,34,60)(13,44,42,64,29,58,41,39)(18,51,49,56,52,57,30,22)(35,62,43,37,63,54,50,55);(1,5,8,6)(2,11,14,12)(3,17,20,19)(4,23,26,24)(7,27,32,28)(9,34,36,15)(10,38,40,21)(13,41,29,42)(16,47,48,53)(18,49,52,30)(22,51,56,57)(25,46,59,31)(33,60,61,45)(35,50,63,43)(37,62,55,54)(39,58,64,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320528, #72320647, #72320677
maximal normal subgroup of	#72320528, #72320647, #72320677
maximal quotient of	#72322850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406612);