#149603

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,20,14,21,32,47,15,22,33,48,35,49,60,69,16,18,34,41,36,42,61,65,12,19,29,44,30,45,57,67)(11,23,25,50,26,51,54,70,24,38,52,46,53,63,71,68,27,43,31,58,55,66,59,72,28,17,37,39,56,40,62,64);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,28)(13,31)(14,26)(15,24)(16,27)(18,38)(19,43)(20,46)(21,40)(22,23)(25,34)(29,52)(30,56)(32,59)(33,37)(35,53)(36,55)(39,41)(42,63)(44,50)(45,66)(47,68)(48,58)(49,51)(54,61)(57,71)(60,62)(64,65)(67,70)(69,72);(10,18)(17,38)(19,22)(20,41)(21,42)(23,43)(39,46)(40,63)(44,48)(45,49)(47,65)(50,58)(51,66)(64,68)(67,69)(70,72)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,d^b=di,e^b=efi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56799);