#149872

order	512
nilpotency class	5
isoclinism	isoclinic class of #149795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#382	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dh,e^b=efhi,f^b=fi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72292967, #72292972
maximal normal subgroup of	#72292967, #72292972

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57068);