#453872

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16182
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

545 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361068);