#152008

order	512
nilpotency class	3
isoclinism	isoclinic class of #1795
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#379
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12679
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

625 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 625 conjugacy classes of subgroups — too many to draw. Only its 144 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=di,f^b=fi,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59204);