#150219

order	512
nilpotency class	5
isoclinism	isoclinic class of #150216
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11692
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2879 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

355 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=gi,e^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efg,g^b=gi,d^c=di,e^d=eh,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, 57415);