#150402

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
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	#391	C₂³
prime radical	C₂×#540	C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 453 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:

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

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,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,d^c=dh,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, 57598);