#150295

order	512
nilpotency class	5
isoclinism	isoclinic class of #150171
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#353
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	C₂×#1352
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,d^2=gh,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dg,e^b=efghi,f^b=fi,h^b=hi,e^d=eh,f^d=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57491);