#127632

order	512
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1519
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8023
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=cgh,d^b=dh,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34828);