#150707

order	512
nilpotency class	5
isoclinism	isoclinic class of #150594
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57903);