#150604

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	#2018	C₂²
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

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

normal subgroups only

This group has 271 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)(3,14,17,44)(4,19,22,51)(6,12,26,42)(7,13,28,41)(8,30,32,52)(9,34,36,21)(10,27,38,23)(15,45,47,62)(16,31,48,40)(18,37,25,56)(20,33,53,60)(24,35,55,61)(29,50,57,64)(39,58,63,54)(43,49,59,46);(1,3)(2,8)(4,20)(5,25)(6,15)(7,29)(9,31)(10,37)(11,40)(12,33)(13,43)(14,39)(16,24)(17,28)(18,21)(19,50)(22,49)(23,52)(26,57)(27,48)(30,58)(32,41)(34,47)(35,45)(36,60)(38,62)(42,59)(44,51)(46,54)(53,55)(56,63)(61,64);(2,9,12,13)(3,15)(5,26)(8,31,33,43)(10,35,19,39)(11,41,42,36)(14,37,45,50)(16,46)(18,29)(20,52)(22,27)(28,34)(32,59,60,40)(38,63,51,61)(44,64,62,56)(55,58)

Polycyclic presentation:

2^9|a^2=e,c^2=hi,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, 57800);