#150603

order	512
nilpotency class	5
isoclinism	isoclinic class of #150602
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11800
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

266 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,11)(3,14,16,45)(4,10,21,37)(6,13,25,43)(7,12,26,41)(8,28,30,19)(9,33,35,20)(15,29,48,40)(17,46,50,60)(18,36,24,55)(22,39,53,42)(23,38,54,63)(27,44,56,64)(31,52,59,49)(32,58,61,51)(34,57,62,47);(1,3,6,17)(2,8,13,32)(4,19,22,51)(5,24,25,56)(7,27,20,18)(9,31,12,29)(10,36,39,44)(11,40,43,59)(14,34,46,38)(15,47,49,23)(16,33,50,26)(21,52,53,48)(28,57,58,54)(30,41,61,35)(37,45,42,60)(55,63,64,62);(2,9,13,12)(5,25)(8,29,32,31)(10,34,39,38)(11,41,43,35)(14,44,46,36)(16,50)(21,53)(24,56)(26,33)(28,58)(30,59,61,40)(37,63,42,62)(45,55,60,64)(48,52)(54,57)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=h,d^2=gi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efg,g^b=gi,e^c=ei,e^d=eh,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57799);