#101624

order	512
nilpotency class	3
isoclinism	isoclinic class of #101519
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

405 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=gh,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egh,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, 8820);