#150188

order	512
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
socle	C₂²	C₂×#353
prime radical	#23421	C₂
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,5)(4,7)(6,8)(9,10,13,18,14,19,29,36)(11,20,22,35,24,31,38,39)(12,16,25,33,26,17,28,34)(15,27,30,21,32,37,40,23);(2,5)(3,8)(6,7)(9,11)(10,15)(12,21)(13,27)(14,24)(16,20)(17,31)(18,35)(19,32)(22,25)(23,26)(28,38)(29,37)(30,34)(33,40)(36,39);(9,12,14,26)(10,16,19,17)(11,21,24,23)(13,25,29,28)(15,20,32,31)(18,33,36,34)(22,37,38,27)(30,35,40,39)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,d^b=dg,e^b=efgi,f^b=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57384);