#94148

order	512
nilpotency class	5
isoclinism	isoclinic class of #7481
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×Q₈	C₄×C₈
Frattini, cosocle	#2931	C₂²
socle	C₂	#7204
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×Q₈ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=gh,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,d^b=df,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, 1344);