#475245

order	512
nilpotency class	3
isoclinism	isoclinic class of #469875
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2607
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17037
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

541 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:

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

Polycyclic presentation:

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

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, 382441);