#150298

order	512
nilpotency class	5
isoclinism	isoclinic class of #150220
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	#390	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2309 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

403 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57494);