#495041

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1942
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

613 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402237);