#471397

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16796
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

529 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,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, 378593);