#149652

order	512
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₈
Frattini, cosocle	#408	C₂³
socle	C₂	C₂×#1326
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

407 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dhi,e^b=ef,h^b=hi,e^d=ehi,f^d=fi,f^e=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, 56848);