#94151

order	512
nilpotency class	5
isoclinism	isoclinic class of #7493
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1390
commutator	C₂×D₄	C₄×C₈
Frattini, cosocle	#2931	C₂²
socle	C₂	#7204
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #2931 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 205 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=g,e^2=gh,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,c^b=ci,d^b=dfi,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=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, 1347);