#473513

order	512
nilpotency class	3
isoclinism	isoclinic class of #456381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16868
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

655 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)(3,14,19,41)(4,9)(5,10,27,39)(6,11)(7,13)(8,18,31,20)(12,15)(16,44,47,32)(17,48)(21,53,46,29)(22,54,55,33)(23,34)(24,36)(25,57,56,37)(26,38,28,40)(30,51)(35,43)(42,52)(45,59)(49,62)(50,60)(58,63)(61,64);(1,3,15,18,7,19,6,20)(2,8,12,14,13,31,11,41)(4,21,23,47,24,46,43,16)(5,17,26,51,27,52,28,50)(9,32,34,29,36,44,35,53)(10,30,38,42,39,60,40,48)(22,49,56,64,55,63,25,45)(33,61,57,58,54,59,37,62);(1,4,6,23,7,24,15,43)(2,9,11,34,13,36,12,35)(3,16,18,46,19,47,20,21)(5,25,26,55,27,56,28,22)(8,29,14,32,31,53,41,44)(10,37,38,54,39,57,40,33)(17,49,50,64,52,63,51,45)(30,59,48,62,60,61,42,58);(1,5,6,28,7,27,15,26)(2,10,11,40,13,39,12,38)(3,17,20,50,19,52,18,51)(4,22,23,56,24,55,43,25)(8,30,41,48,31,60,14,42)(9,33,34,57,36,54,35,37)(16,45,21,49,47,64,46,63)(29,58,44,59,53,62,32,61)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380709);