#94422

order	512
nilpotency class	5
isoclinism	isoclinic class of #94419
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7304
commutator	C₂²×Q₈	C₄²
Frattini, cosocle	C₂×#536	C₂²
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂×#536 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 44 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=df,h^b=hi,d^c=dhi,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1618);