#94162

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#354
commutator	C₄²	C₄×C₈
Frattini, cosocle	#2931	C₂²
socle	C₂	#7205
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #2931 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

201 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=g,e^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ci,d^b=dfh,d^c=dhi,e^d=ei,f^d=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, 1358);