#150728

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#23941	C₂
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2311 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 436 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,d^c=dh,e^c=ei,f^c=fi,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, 57924);