#149995

order	512
nilpotency class	5
isoclinism	isoclinic class of #149931
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#384	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2326 ◁ G
LECS	G ▷ #384 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 292 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gi,d^2=hi,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgh,d^b=dhi,e^b=efh,f^b=fi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57191);