#150891

order	512
nilpotency class	5
isoclinism	isoclinic class of #150397
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

260 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,4)(7,8)(9,10,13,19,15,21,29,38)(11,22,24,37,12,17,26,35)(14,18,31,36,23,20,30,39)(16,32,33,25,34,28,40,27);(1,2,5,8)(3,6,4,7)(9,11,23,27,15,12,14,25)(10,16,18,22,21,34,20,17)(13,28,31,26,29,32,30,24)(19,37,39,33,38,35,36,40);(1,3,5,4)(2,7,8,6)(9,12,15,11)(10,17,21,22)(13,26,29,24)(14,25,23,27)(16,18,34,20)(19,35,38,37)(28,30,32,31)(33,36,40,39)

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=gi,d^2=gh,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efghi,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, 58087);