#100417

order	512
nilpotency class	3
isoclinism	isoclinic class of #100251
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

520 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=hi,d^2=gh,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7613);