#150369

order	512
nilpotency class	5
isoclinism	isoclinic class of #150220
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

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

normal subgroups only

This group has 253 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=hi,c^2=ghi,d^2=gi,e^2=i,f^2=i,b^a=bd,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^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, 57565);