#93913

order	512
nilpotency class	5
isoclinism	isoclinic class of #7410
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1344
commutator	C₂²×C₈	C₂×C₈
Frattini, cosocle	#1729	C₂²
prime radical	C₂⁴	#104
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ #1729 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=f,d^2=g,f^2=i,b^a=bc,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=defi,f^b=fi,d^c=dh,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1109);