#433920

order	512
nilpotency class	3
isoclinism	isoclinic class of #433919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15493
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341116);