#94141

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, socle	C₂²	#1390
commutator	C₂×Q₈	C₄×C₈
Frattini, cosocle	C₄×C₄⋊Q₈	C₂²
prime radical	C₂³	#342
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄×C₄⋊Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

229 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,4,7,6,8,5,3)(9,10)(11,19,21,35,23,36,22,37)(12,24)(13,27)(14,17)(15,20,30,38,31,40,32,39)(16,25)(18,29)(26,34)(28,33);(1,3,5,8,6,7,4,2)(9,11)(10,15,17,31)(12,20)(13,21)(14,23)(16,19,33,36)(18,30,27,32)(22,29)(24,35,34,37)(25,38)(26,40)(28,39)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=ghi,f^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, 1337);