#99696

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#386
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

397 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=gi,d^2=g,e^2=hi,f^2=gh,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6892);